Hopf algebras of primitive Lie pseudogroups and Hopf cyclic cohomology

Hopf algebras of primitiv e Lie pseudo groups and Hopf cyclic cohomolo gy Henri Mosco vici ∗ † and Bahram Rangip our ‡ Abstract W e asso ciate to each inﬁnite primitive Lie pseudogroup a Hopf algebra of ‘transverse symmetries ’, b y reﬁning a procedur e due to Connes and the ﬁr s t author in the case of the general pseudogroup. The a ﬃliated Hopf alge bra can b e viewed as a ‘quantum group’ counterpart of the inﬁnite-dimensional primitive Lie algebra of the ps e udogroup. It is ﬁrst cons tructed via its action on the ´ etale group oid a sso ciated to the pseudogro up, and then realized as a bicr ossed pro duct of a universal env eloping algebra by a Ho pf algebr a of regular functions on a for - mal group. The bicr ossed pro duct structure allows to ex pr ess its Hopf cyclic cohomolo gy in terms o f a bico cyclic bicomplex ana logous to the Chev alley-Eilenberg complex. As an application, we compute the r el- ative Hopf cyclic c ohomolog y mo dulo the linear iso tropy for the Hopf algebra o f the genera l pseudogroup, and ﬁnd explicit co cycle re presen- tatives for the univ ers al Chern classes in Hopf cyclic co homology . As another application, we determine all Ho pf cyclic cohomolog y groups for the Hopf algebr a asso ciated to the pseudog r oup of loc a l diﬀeomor- phisms of the line. In tro duction The transverse charact eristic classes of foliations w ith holonom y i n a t ran- sitiv e Lie pseu dogroup Γ of lo cal diﬀeomorphisms of R n are most eﬀectiv el y describ ed in the f ramew ork of the Gelfand-F uks [11] cohomology of the Lie ∗ Researc h supp orted by the National Science F ound ation aw ard no. D MS-0245481. † Department of Mathematics, Colum bus, OH 43210, USA ‡ Department of Mathematics and Statistics, Univ ersity of New Brunswick, F rederic ton, NB, Canada 1 algebra of formal v ector ﬁelds associated to Γ ( cf. e.g. Bo tt-Haeﬂiger [2]). In the dual, K -homologica l con text, the transv erse c haracteristic c lasses of general foliatio ns ha v e b een expressed by Connes and the ﬁrst a uthor ( cf. [6, 7, 8]) in terms of the Hopf cyclic cohomology of a Hopf algebra H n canonically asso ciated to the group Diﬀ R n . In t his pap er w e constru ct s im ilar Hopf algebras for all classical groups of diﬀeomorphisms, or equ iv alen tly for the in ﬁnite p rimitiv e Lie-Cartan pseu- dogroups [3] of lo cal C ∞ -diﬀeomorphisms. The Hopf alg ebra H Π asso ciated to suc h a pseu d ogroup Π can be regarded as a ‘quan tum group’ analog of the in ﬁ nite dimensional p rimitiv e Lie algebra of Π ( cf. Singer-Stern b erg [26 ], Guillemin [13]). It is initially constructed via its tautologi cal ac tion on the ´ etale group oid asso ciated to Π, and is then reconstructed, in a mann er rem- iniscen t of a ‘quantum double’, as the bicrossed pr o duct of a univ ersal en - v eloping algebra by a Hopf algebra of regular fu nctions on a form al group . In turn, the b icrossed pro d uct structure is emplo y ed to redu ce the compu- tation of the Hopf cyclic cohomolog y of H Π to that a bico cyclic bicomplex analogous to th e Chev alley-Eilen b erg complex. This apparatus is then ap- plied to compute the relativ e Hopf cyclic cohomology of H n mo dulo gl n . W e actually ﬁ nd explicit co cycles repr esen ting the Hopf cyclic analog ues of th e unive rsal Chern classes. In the case of H 1 , the improv ed tec hniqu e allo ws us to reﬁne our previous compu tations [24] and completely determine the non- p erio dized Hopf cyclic cohomology of the Hopf algebra H 1 aﬃliated with the pseu d ogroup of lo cal diﬀeomorph ism s of the line. W e no w giv e a brief outline of the main results. O ur construction of the Ho pf algebra asso ciated to a primitiv e Lie pseudogroup Π is mo d eled on that of H n in [6], wh ich in tu rn w as insp ired by a pro cedur e du e to G. I. Kac [ 18] for prod ucing non-comm utativ e and non-co commutat ive qu an tum group s out of ‘matc hed pairs’ of ﬁnite groups. It relies on splitting the group Diﬀ Π of globally deﬁn ed diﬀeomorphisms of t yp e Π as a set-theoretical pro duct of t w o subgroups, Diﬀ Π = G Π · N Π , G Π ∩ N Π = { e } . F or a ﬂat ( i.e. con taining all the translations) primitiv e pseudogroup Π, G Π is the subgroup consisting of the aﬃne transformations of R n that are in Diﬀ Π , while N Π is the subgroup consisting of those diﬀeomorphisms in Diﬀ Π that preserve the origin to order 1. The pseudogroup of con tact tr an s formations is the only inﬁnite primitiv e pseudogroup which is not ﬂat. In that case, we identify R 2 n +1 with the Heisen b erg g roup H n . Instead of the vec tor translations, we let H n act on 2 itself b y group left translations, and deﬁne the group of ‘aﬃne Heisen b erg transformations’ G Π as the semidirect pro du ct of H n b y the linear isotropy group G 0 Π consisting of the linear conta ct transformations. As factor N Π w e tak e the subgroup of all conta ct diﬀeomorphisms preservin g the origin to order 1 in the sense of Heisenberg calculus, i. e . whose diﬀeren tial at 0 is th e iden tit y map of the Heisenber g tangen t b undle ( cf. e.g. [25]). The factorizatio n Diﬀ Π = G Π · N Π allo ws to r epresen t u niquely any φ ∈ Diﬀ Π as a pro duct φ = ϕ · ψ , with ϕ ∈ G Π and ψ ∈ N Π . F actorizing the pro du ct of an y tw o elements ϕ ∈ G Π and ψ ∈ N Π in the reverse order, ψ · ϕ = ( ψ ⊲ ϕ ) · ( ψ ⊳ ϕ ), one obtains a left action ψ 7→ ˜ ψ ( ϕ ) := ψ ⊲ ϕ of N Π on G Π , along with a righ t acti on ⊳ of G Π on N Π . Equiv alen tly , these actions are restrictions of the natural actions of Diﬀ Π on the coset s p aces Diﬀ Π /N Π ∼ = G Π and G Π \ Diﬀ Π ∼ = N Π . The ‘d ynamical’ deﬁnition of the Hopf algebra H Π asso ciated to the pseu- dogroup Π is obtained by means of its act ion on the (d iscrete) crossed pro d - uct algebra A Π = C ∞ ( G Π ) ⋊ Diﬀ Π , wh ic h arises as follo ws. O ne starts with a ﬁ xed basis { X i } 1 ≤ i ≤ m for the Lie algebra g Π of G Π . Eac h X ∈ g Π giv es rise to a left-in v arian t vecto r ﬁ eld on G Π , whic h is th en extended to a linear op erator on A Π , in the most ob vious fashion: X ( f U φ − 1 ) = X ( f ) U φ − 1 , where f ∈ C ∞ ( G Π ) and φ ∈ Diﬀ Π . O ne has U φ − 1 X i U φ = m X j =1 Γ j i ( φ ) X j , i = 1 , . . . , m, with Γ j i ( φ ) ∈ C ∞ ( G cn ), and we deﬁne corresp onding m ultiplicat ion op era- tors on A Π b y taking ∆ j i ( f U φ − 1 ) = ( Γ ( φ ) − 1 ) j i f U ∗ φ , where Γ ( φ ) =  Γ j i ( φ )  1 ≤ i,j ≤ m . As an algebra, H Π is generated b y the op erators X k ’s and ∆ j i ’s. In partic- ular, H Π con tains all iterated comm utators ∆ j i,k 1 ...k r := [ X k r , . . . , [ X k 1 , ∆ j i ] . . . ] , whic h are m ultiplication op erators b y the functions Γ j i,k 1 ...k r ( φ ) := X k r . . . X k 1 (Γ j i ( φ )) , φ ∈ Diﬀ Π . 3 F or an y a, b ∈ A (Π cn ), one has X k ( ab ) = X k ( a ) b + X j ∆ j k ( a ) X j ( b ) , ∆ j i ( ab ) = X k ∆ k i ( a ) ∆ j k ( b ) , and by multiplicati vit y ev ery h ∈ H (Π cn ) satisﬁes a ‘Leibniz rule’ of the form h ( ab ) = X h (1) ( a ) h (2) b ) , ∀ a, b ∈ A Π . The op erators ∆ • •···• satisfy the follo wing Bianc hi-t yp e ident ities: ∆ k i,j − ∆ k j,i = X r,s c k r s ∆ r i ∆ s j − X ℓ c ℓ ij ∆ k ℓ , where c i j k are the structur e constan ts of the Lie algebra g Π . Theorem 0.1. L et H Π b e the abstr act Lie algebr a gener ate d b y the op er ators { X k , ∆ j i,k 1 ...k r } and their c omm utation r elations. 1. The algebr a H Π is isomorp hic to the q uotient of the universal envelop- ing algebr a U ( H Π ) by the ide al B Π gener at e d by the Bianchi identities. 2. The L eibniz rule determines u ni q uely a c opr o duct, with r esp e ct to which H Π a Hopf algebr a and A Π an H Π -mo dule algebr a. W e next describ e the b icr os se d pr o duct realizatio n of H Π . Let F Π denote the algebra of functions on N generated by the jet ‘co ordinates’ η j i,k 1 ...k r ( ψ ) := Γ j i,k 1 ...k r ( ψ )( e ) , ψ ∈ N ; the deﬁnition is ob viously ind ep endent of the choice of basis for g Π . F ur- thermore, F Π is a Hopf algebra with copro duct uniquely and well-deﬁned b y the rule ∆ f ( ψ 1 , ψ 2 ) := f ( ψ 1 ◦ ψ 2 ) , ∀ ψ 1 , ψ 2 ∈ N , and with antipo de S f ( ψ ) := f ( ψ − 1 ) , ψ ∈ N . No w the univ ersal env elo ping alg ebra U Π = U ( g Π ) can be equipp ed with a righ t F Π -comod ule co algebra structure H : U Π → U Π ⊗ F Π as fol lo ws. Let 4 { X I = X i 1 1 · · · X i m m ; i 1 , . . . , i m ∈ Z + } b e the PBW basis o f U Π induced b y the c hosen basis of g Π . T hen U ψ − 1 X I U ψ = X J β J I ( ψ ) X J , with β J I ( ψ ) in the algebra of functions on G Π generated b y Γ j i,K ( ψ ). O ne obtains a coaction H : U Π → U Π ⊗ F Π b y deﬁn ing H ( X I ) = X J X J ⊗ β J I ( · )( e ); again, the deﬁnition is indep enden t of the choic e of basis. The righ t action ⊳ of G Π on N Π induces an action of G Π on F Π and h ence a left action of U Π on F Π , that m akes F Π a left U Π -mo dule algebra. Theorem 0.2. With the ab ove op er ations, U Π and F Π form a matche d p air of Hopf algebr as, and their bicr osse d pr o duct F Π ◮ ⊳ U Π is c anonic al ly isomorph ic to the Hopf algebr a H cop Π . The Hopf algebra H Π also comes equipp ed with a mo dular c haracter δ = δ Π , extending the in ﬁ nitesimal modular c haracter δ ( X ) = T r(ad X ), X ∈ g Π . The corresp onding mo du le C δ , viewed also as a trivial como du le, d eﬁ nes a ‘mo d ular pair in inv olution’, cf. [7], or a particular case of an ‘S A YD mo dule-como dule’, cf. [16]. Suc h a datum allo ws to sp ecialize Connes’ E xt Λ -deﬁnition [4] of cyclic cohomology to the conte xt of Hopf alge bras ( cf. [6, 7]). The r esulting Hopf cyclic cohomology , in tro duced in [6] and extended to SA YD co eﬃcients in [17], incorp orates b oth Lie algebra and group cohomology and pr o vides the appropriate cohomolog ical tool for the treatmen t of symm etry in noncommutativ e geometry . Ho w ev er, its com- putation for general, i.e. non-commutat ive and non-co comm utativ e, Hopf algebras p oses quite a c hallenge. In the case of H n , it has b een sho wn in [6] that H P ∗ ( H n ; C δ ) is canonically isomo rp hic to the Gelfand-F uks co homol- ogy of the Lie algebra a n of formal ve ctor ﬁelds on R n , result wh ic h al lo wed to transfer th e transv erse c h aracteristic classes of foliations from K -theory classes into K -homology c haracteristic cla sses. There are ve ry few instances of direct calculatio ns so f ar (see e . g. [24], where the p erio d ic Hopf cyclic cohomology of sev eral v arian ts of H 1 has b een directly compu ted), and a general mac hinery for p erforming suc h computations is only b eginning to emerge. W e rely on the bicrossed pro d uct structure of the Hopf alge bra H Π to reduce the compu tation of its Hopf cyclic cohomology to that of a simpler, bicocyclic 5 bicomplex. T h e latter com bines the Chev alley-Eile nber g complex of the Lie algebra g = g Π with coeﬃcien ts in C δ ⊗ F ⊗• and the coalgebra cohomology complex of F = F Π with co eﬃcien ts in ∧ • g : . . . ∂ g   . . . ∂ g   . . . ∂ g   C δ ⊗ ∧ 2 g β F / / ∂ g   C δ ⊗ F ⊗ ∧ 2 g β F / / ∂ g   C δ ⊗ F ⊗ 2 ⊗ ∧ 2 g β F / / ∂ g   . . . C δ ⊗ g β F / / ∂ g   C δ ⊗ F ⊗ g β F / / ∂ g   C δ ⊗ F ⊗ 2 ⊗ g ∂ g   β F / / . . . C δ ⊗ C β F / / C δ ⊗ F ⊗ C β F / / C δ ⊗ F ⊗ 2 ⊗ C β F / / . . . . Theorem 0.3. 1. The ab ove bic omp lex c omputes the p e rio dic Hopf c yclic c oh omolo gy H P ∗ ( H Π ; C δ ) . 2. Ther e is a r elative version of the ab ove bic omp lex that c omputes the r e lative p erio dic Hopf cyclic c ohomol o gy H P ∗ ( H Π , U ( h ); C δ ) , for any r e ductive sub algebr a h of the line ar isotr opy Lie algebr a g 0 Π . As the main applicati on in this pap er, w e compute the p erio dic Hopf cycl ic cohomology of H n relativ e to gl n and ﬁn d exp licit co cycle repr esen tativ es for its basis, as describ ed b elo w. F or eac h partition λ = ( λ 1 ≥ . . . ≥ λ k ) of th e set { 1 , . . . , p } , wh ere 1 ≤ p ≤ n , w e let λ ∈ S p also den ote a permutation whose cycles ha v e lengths λ 1 ≥ . . . ≥ λ k , i.e. represen ting the corresp on d ing conjugacy class [ λ ] ∈ [ S p ]. W e then deﬁn e C p,λ := X ( − 1) µ 1 ⊗ η j 1 µ (1) ,j λ (1) ∧ · · · ∧ η j p µ ( p ) ,j λ ( p ) ⊗ X µ ( p +1) ∧ · · · ∧ X µ ( n ) , where the s ummation is o v er all µ ∈ S n and all 1 ≤ j 1 , j 2 , . . . , j p ≤ n . Theorem 0.4. The c o chains { C p,λ ; 1 ≤ p ≤ n, [ λ ] ∈ [ S p ] } ar e c o cycles and their classes f orm a b asis of the gr oup H P ǫ ( H n , U ( gl n ); C δ ) , wher e ǫ ≡ n mo d 2 , while H P 1 − ǫ ( H n , U ( gl n ); C δ ) = 0 . The corresp on d ence with the u niv ersal Chern classes is ob vious. Let P n [ c 1 , . . . c n ] = C [ c 1 , . . . c n ] / I n 6 denote th e truncated p olynomial rin g, where deg( c j ) = 2 j , and I n is the ideal generated b y the monomials of degree > 2 n . T o eac h partition λ as ab o v e, one asso ciates th e degree 2 p mon omial c p,λ := c λ 1 · · · c λ k , λ 1 + . . . + λ k = p ; the corresp ond in g classes { c p,λ ; 1 ≤ p ≤ n, λ ∈ [ S p ] } form a basis of the v ector sp ace P n [ c 1 , . . . c n ]. A second application is the complete determination of the Ho c hsc hild co- homology o f H 1 and of the n on -p erio dized Ho pf cyclic cohomology groups H C q ( H 1 ; C δ ), where δ is the mo dular characte r of the Hopf algebra H 1 . Con ten ts 1 Construction via Hopf actions 7 1.1 Hopf algebra of th e general pseudogroup . . . . . . . . . . . . 8 1.2 The general case of a ﬂat primitive pseudogroup . . . . . . . 21 1.2.1 Hopf algebra of th e volume pr eservin g pseudogroup . 23 1.2.2 Hopf algebra of th e s y m plectic pseu d ogroup . . . . . . 24 1.3 Hopf algebra of th e con tact pseud ogroup . . . . . . . . . . . . 25 2 Bicrossed pro duct realization 38 2.1 The ﬂat case . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.2 The non-ﬂat case . . . . . . . . . . . . . . . . . . . . . . . . . 52 3 Hopf cy clic cohomolog y 54 3.1 Quic k synops is of Hopf cyclic cohomology . . . . . . . . . . . 54 3.2 Reduction to diagonal m ixed complex . . . . . . . . . . . . . 61 3.3 Bicocyclic complex for p rimitiv e Hopf algebras . . . . . . . . 78 3.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.4.1 Hopf cyclic Ch ern classes . . . . . . . . . . . . . . . . 83 3.4.2 Non-p erio dized Hopf cyclic cohomology of H 1 . . . . . 89 1 Construction via Hopf actions This section is dev oted to the ‘dyn amical’ co nstr u ction of the Hopf algebras asso ciated to primitive Lie pseudogroup s of inﬁnite type. F or the sak e of clarit y , we start with the case of the general pseu dogroup ( cf. also [9]), where the te c hnical details can b e handled in the most t ransp aren t f ashion. T he 7 other cases of ﬂat pseudogroups can b e treated in a similar manner. In order to illus trate th e sligh t adju s tmen ts needed to co v er them, w e describ e in some detail the Hopf algebras aﬃliated to the v olume p reserving and the symplectic pseud ogroups. On the other hand , the case of the cont act pseudogroup, whic h is the only non-ﬂat one, requ ires a certain c hange of geometric viewp oint, namely replacing th e natural motions of R 2 n +1 with the natural motions of the Heisen b erg group H n . 1.1 Hopf algebra of t he general pseudogroup Let F R n → R n b e the fr ame b undle on R n , which we identify to R n × GL( n, R ) in the ob vious w a y: the 1-jet at 0 ∈ R n of a germ of a lo cal diﬀeomorphism φ on R n is view ed as the pair  x := φ (0) , y := φ ′ 0 (0)  ∈ R n × GL( n, R ) , φ 0 ( x ) := φ ( x ) − φ (0) . (1.1) The ﬂat connection on F R n → R n is given by the matrix-v al ued 1-form ω = ( ω i j ) wh ere, with the usu al summation con v ent ion, ω i j := ( y − 1 ) i µ d y µ j = ( y − 1 d y ) i j , i, j = 1 , . . . , n , (1.2) and the canonical form is the v ector-v alued 1-form θ = ( θ k ) , θ k := ( y − 1 ) k µ dx µ = ( y − 1 dx ) k , k = 1 , . . . , n (1.3) The basic h orizon tal vec tor ﬁelds for the ab o v e conn ection are X k = y µ k ∂ µ , k = 1 , . . . , n , wh ere ∂ µ = ∂ ∂ x µ , (1.4) and the fu ndament al ve rtical v ector ﬁelds asso ciated to the standard b asis of gl ( n, R ), formed by the elemen tary matrices { E j i ; 1 ≤ i, j ≤ n } , ha v e the expression Y j i = y µ i ∂ j µ , i, j = 1 , . . . , n , where ∂ j µ := ∂ ∂ y µ j . (1.5) Let G := R n ⋊ GL( n, R ) denote the group of aﬃne motions of R n . Prop osition 1.1. The ve ctor ﬁelds { X k , Y j i ; i, j, k = 1 , . . . , n } form a b asis of left-invariant ve ctor ﬁelds on the g r oup G . 8 Pr o of. Represent G as the s u bgroup of GL( n + 1 , R ) consisting of the ma- trices a =  y x 0 1  , with y ∈ GL( n, R ), a nd x ∈ R n . L et { e k , E j i ; i, j, k = 1 , . . . , n } b e the standard basis of the Lie algebra g := R n ⋊ gl ( n, R ), and denote by { ˜ e k , ˜ E j i ; i, j, k = 1 , . . . , n } the corresp onding left- inv arian t v ector ﬁelds. By deﬁnition, at the p oint a , ˜ e k is tangen t to the curve t 7→  y x 0 1   1 te k 0 1  =  y t y e k + x 0 1  and therefore coincides with X k = P µ y µ k ∂ µ , while ˜ E j i is tangen t to t 7→  y x 0 1  e tE j i 0 0 1 ! = y e tE j i 0 0 1 ! whic h is precisely Y j i = P µ y µ i ∂ j µ . The group of diﬀeomorphisms G := Diﬀ R n acts on F R n , by the n atural lift of the tautological actio n to the frame lev el: e ϕ ( x, y ) :=  ϕ ( x ) , ϕ ′ ( x ) · y  , wh ere ϕ ′ ( x ) i j = ∂ j ϕ i ( x ) . (1.6) Viewing here G as a discrete group, w e form the crossed pro d uct alge bra A := C ∞ c ( F R n ) ⋊ G . As a v ector space, it is spanned by monomials of the form f U ∗ ϕ , where f ∈ C ∞ c ( F R n ) and U ∗ ϕ stands for e ϕ − 1 , while the pro duct is giv en by t he m ultiplication rule f 1 U ∗ ϕ 1 · f 2 U ∗ ϕ 2 = f 1 ( f 2 ◦ e ϕ 1 ) U ∗ ϕ 2 ϕ 1 . (1.7) Alternativ ely , A can b e regarded as the subalgebra of the en domorphism algebra L ( C ∞ c ( F R n )) = En d C  C ∞ c ( F R n )  , generated by the multiplicatio n and the tr anslation op erators M f ( ξ ) = f ξ , f ∈ C ∞ c ( F R n ) , ξ ∈ C ∞ c ( F R n ) (1.8) U ∗ ϕ ( ξ ) = ξ ◦ e ϕ , ϕ ∈ G , ξ ∈ C ∞ c ( F R n ) . (1.9) Since the right act ion of GL( n, R ) on F R n comm utes with the action of G , at the Lie algebra lev el one has U ϕ Y j i U ∗ ϕ = Y j i , ϕ ∈ G . (1.10) 9 This allo ws to promote the v ertical v ector ﬁ elds to deriv ations of A . Indeed, setting Y j i ( f U ∗ ϕ ) = Y j i ( f ) U ∗ ϕ , f U ∗ ϕ ∈ A , (1.11) the extended op erators satisfy th e deriv ati on rule Y j i ( a b ) = Y j i ( a ) b + a Y j i ( b ) , a, b ∈ A , (1.12) W e also prolong the horizon tal vecto r ﬁ elds to linear transf orm ations X k ∈ L ( A ), in a similar fash ion: X k ( f U ∗ ϕ ) = X k ( f ) U ∗ ϕ , f U ∗ ϕ ∈ A . (1.13) The resulting op erators are no longer G -inv arian t. Instead of (1.10), they satisfy U ∗ ϕ X k U ϕ = X k − γ i j k ( ϕ ) Y j i , (1.14) where γ i j k ( ϕ )( x, y ) =  y − 1 · ϕ ′ ( x ) − 1 · ∂ µ ϕ ′ ( x ) · y  i j y µ k . ( 1.15) Using the left- inv ariance of the vecto r ﬁelds X k and (1.14), or just the explicit form ula (1.15), one sees that ϕ 7→ γ i j k ( ϕ ) is a group 1-cocycle on G with v al ues in C ∞ ( F R n ); sp eciﬁcally , γ i j k ( ϕ ◦ ψ ) = γ i j k ( ϕ ) ◦ ˜ ψ + γ i j k ( ψ ) , ∀ ϕ, ψ ∈ G . (1.16) As a consequ ence of (1.14), the op erators X k ∈ L ( A ) are no longer deriv a- tions of A , but satisfy instead a non-symmetric Leibniz rule: X k ( a b ) = X k ( a ) b + a X k ( b ) + δ i j k ( a ) Y j i ( b ) , a, b ∈ A , (1.17) where the lin ear op erators δ i j k ∈ L ( A ) are deﬁn ed b y δ i j k ( f U ∗ ϕ ) = γ i j k ( ϕ ) f U ∗ ϕ . (1.18) Indeed, on taking a = f 1 U ∗ ϕ 1 , b = f 2 U ∗ ϕ 2 , one has X k ( a · b ) = X k ( f 1 U ∗ ϕ 1 · f 2 U ∗ ϕ 2 ) = X k ( f 1 · U ∗ ϕ 1 f 2 U ϕ 1 ) U ∗ ϕ 2 ϕ 1 = X k ( f 1 ) U ∗ ϕ 1 · f 2 U ∗ ϕ 2 + f 1 U ∗ ϕ 1 · X k ( f 2 U ∗ ϕ 2 ) + f 1 U ∗ ϕ 1 · ( U ϕ 1 X k U ∗ ϕ 1 − X k )( f 2 U ∗ ϕ 2 ) , whic h together with (1.14) and the cocycle prop ert y (1.16) imply (1.17). 10 The same co cycle prop erty sho ws that the op erators δ i j k are deriv ations: δ i j k ( a b ) = δ i j k ( a ) b + a δ i j k ( b ) , a, b ∈ A , (1.19) The op erators { X k , Y i j } satisfy the comm utation relations of the group of aﬃne transform ations of R n : [ Y j i , Y ℓ k ] = δ j k Y ℓ i − δ ℓ i Y j k , (1.20) [ Y j i , X k ] = δ j k X i , [ X k , X ℓ ] = 0 . The su ccessiv e commutato rs of the op erators δ i j k ’s with the X ℓ ’s yield new generations of δ i j k ℓ 1 ...ℓ r := [ X ℓ r , . . . [ X ℓ 1 , δ i j k ] . . . ] , (1.21) whic h inv olv e multiplicatio n by higher order jets of diﬀeomorphisms δ i j k ℓ 1 ...ℓ r ( f U ∗ ϕ ) := γ i j k ℓ 1 ...ℓ r ( ϕ ) f U ∗ ϕ , where (1.22) γ i j k ℓ 1 ...ℓ r ( ϕ ) := X ℓ r · · · X ℓ 1  γ i j k ( ϕ )  . Eviden tly , they comm ute among themselv es: [ δ i j k ℓ 1 ...ℓ r , δ i ′ j ′ k ′ ℓ ′ 1 ...ℓ r ] = 0 . (1.2 3) The comm utators b et we en the Y λ ν ’s and δ i j k ’s, whic h can b e easily obtained from the exp licit expression (1.15 ) of the co cycle γ , are as follo ws: [ Y λ ν , δ i j k ] = δ λ j δ i ν k + δ λ k δ i j ν − δ i ν δ λ j k . More generally , one chec ks by induction the relations [ Y λ ν , δ i j 1 j 2 j 3 ...j r ] = r X s =1 δ λ j s δ i j 1 j 2 j 3 ...j s − 1 ν j s +1 ...j r − δ i ν δ λ j 1 j 2 j 3 ...j r . (1.24) The comm utator r elations (1 .20), (1.21), (1.23), (1.24) sho w that the s u b- space h n of L ( A ) generated b y the op erators { X k , Y i j , δ i j k ℓ 1 ...ℓ r ; i, j, k , ℓ 1 . . . ℓ r = 1 , . . . , n, r ∈ N } (1.25) forms a Lie algebr a . W e let H n denote the unital sub algebr a of L ( A ) generated by h n . Unlik e the co dimension 1 case (cf. [6 ]), H n do es not coincide with the unive rsal en v eloping alge bra A ( h n ) w h en n ≥ 2. Indeed, ﬁrst of all th e op erators 11 δ i j k ℓ 1 ...ℓ r are not all distinct; the order of th e ﬁrst tw o lo w er indices or of the last r indices is immaterial. In deed, the expr ession of the co cycle γ , γ i j k ( ϕ )( x, y ) = ( y − 1 ) i λ ( ϕ ′ ( x ) − 1 ) λ ρ ∂ µ ∂ ν ϕ ρ ( x ) y ν j y µ k , (1.26) is clearly symmetric in the indices j and k . The sy m metry in the last r indices follo ws from the deﬁnition (1.21) and the fact that, the connection b eing ﬂat, the horizon tal v ecto r ﬁelds comm ute. It ca n also b e directly seen from the exp licit form ula γ i j k ℓ 1 ...ℓ r ( ϕ )( x, y ) = (1.27) = ( y − 1 ) i λ ∂ β r . . . ∂ β 1  ( ϕ ′ ( x ) − 1 ) λ ρ ∂ µ ∂ ν ϕ ρ ( x )  y ν j y µ k y β 1 ℓ 1 . . . y β r ℓ r . Prop osition 1.2. The op e r ators δ • •···• satisfy the identities δ i j ℓ k − δ i j k ℓ = δ s j k δ i sℓ − δ s j ℓ δ i sk . (1.28) Pr o of. These Bi anc hi-type identi ties are an expr ession of the f act that the underlying connection is ﬂat. Ind eed, if w e let a, b ∈ A and apply (1.17) we obtain X ℓ X k ( a b ) = X ℓ X k ( a ) b + X k ( a ) X ℓ ( b ) + δ i j ℓ ( X k ( a )) Y j i ( b ) + X ℓ ( a ) X k ( b ) + a X ℓ X k ( b ) + δ i j ℓ ( a ) Y j i ( X k ( b )) + X ℓ ( δ i j k ( a )) Y j i ( b ) + δ i j k ( a ) X ℓ ( Y j i ( b )) + δ r sℓ ( δ i j k ( a )) Y s r ( Y j i ( b )) . Since [ X k , X ℓ ] = 0, by an tisymmetrizing in k , ℓ it follo ws that δ i j ℓ ( X k ( a )) Y j i ( b ) + δ i j ℓ ( a ) Y j i ( X k ( b )) + X ℓ ( δ i j k ( a )) Y j i ( b ) + δ i j k ( a ) X ℓ ( Y j i ( b )) + δ r sℓ ( δ i j k ( a )) Y s r ( Y j i ( b )) = δ i j k ( X ℓ ( a )) Y j i ( b ) + δ i j k ( a ) Y j i ( X ℓ ( b )) + X k ( δ i j ℓ ( a )) Y j i ( b ) + δ i j ℓ ( a ) X k ( Y j i ( b )) + δ r sk ( δ i j ℓ ( a )) Y s r ( Y j i ( b )) . Using the ‘aﬃne’ r elations (1.20) and the s y m metry of δ i j k in the low er indices one readily obtains the equation (1.28). In view of this result, the algebra H n admits a basis similar to the P oincar ´ e- Birkhoﬀ-Witt b asis of a universal en v eloping algebra. The n otation needed 12 to sp ecify suc h a basis inv olv es tw o kinds of m ulti-indices. The ﬁ rst kind are of th e form I = n i 1 ≤ . . . ≤ i p ;  j 1 k 1  ≤ . . . ≤  j q k q o , (1.29) while the second kind are of the form K = { κ 1 ≤ . . . ≤ κ r } , where κ s =  i s j s ≤ k s ≤ ℓ s 1 ≤ . . . ≤ ℓ s p s  , s = 1 , . . . , r ; (1.30) in b oth cases the in n er multi-indices are ordered lexicographically . W e then denote Z I = X i 1 . . . X i p Y j 1 k 1 . . . Y j q k q and δ K = δ i 1 j 1 k 1 ℓ 1 1 ...ℓ 1 p 1 . . . δ i r j r k r ℓ r 1 ...ℓ r p r . (1.31) Prop osition 1.3. The monomials δ K Z I , or der e d lexic o gr ap hic al ly, form a line ar b asis of H n . Pr o of. W e need to p ro v e th at if c I ,κ ∈ C are such that X I ,K c I ,K δ K Z I ( a ) = 0 , ∀ a ∈ A , (1.32) then c I ,K = 0, for an y ( I , K ). T o this end , we ev aluate (1.32) on all elemen ts of the form a = f U ∗ ϕ at the p oint e = ( x = 0 , y = I ) ∈ F R n = R n × GL( n, R n ) . In particular, for an y ﬁ xed but arbitrary ϕ ∈ G , one obtains X I X K c I ,K γ κ ( ϕ )( e ) ! ( Z I f )( e ) = 0 , ∀ f ∈ C ∞ c ( F R n ) . (1.33) Since the Z I ’s form a PBW basis of A ( R n ⋊ gl ( n, R )), whic h can b e view ed as the algebra of left-in v arian t diﬀerential op erators on F R n , th e v alidit y of (1.33) for an y f ∈ C ∞ c ( F R n ) implies the v anishing for eac h I of the corresp ondin g coeﬃcient. One th erefore obtains, f or an y ﬁxed I , X K c I ,K γ K ( ϕ )( e ) = 0 , ∀ ϕ ∈ G . (1.34 ) 13 T o p ro v e the v anishing of all the co eﬃcient s, w e shall use induction on the height of K = { κ 1 ≤ . . . ≤ κ r } ; the latter is d eﬁned b y coun ting the total n umb er of horizon tal deriv a tiv es of its largest comp onen ts: | K | = ℓ r 1 + · · · + ℓ r p r . W e start with the case of heigh t 0, when the iden tit y (1.34) reads X K c I ,K γ i 1 j 1 k 1 ( ϕ )( e ) · · · γ i r j r k r ( ϕ )( e ) = 0 , ∀ ϕ ∈ G . Let G 0 b e the subgroup of all ϕ ∈ G such that ϕ (0) = 0. Cho osing ϕ in the subgroup G (2) (0) ⊂ G 0 consisting of t he diﬀeomorphisms whose 2-jet at 0 is of the form J 2 0 ( ϕ ) i ( x ) = x i + 1 2 n X j,k =1 ξ i j k x j x k , ξ ∈ R n 3 , ξ i j k = ξ i k j , and using (1.26), one obtains: X K c I ,K ξ i 1 j 1 k 1 · · · ξ i r j r k r = 0 , ξ i j k ∈ R n 3 , ξ i j k = ξ i k j . It follo ws that all co eﬃcients c I ,K = 0. Let n o w N ∈ N b e th e largest heig ht of occurring in (1.34). By v arying ϕ in the sub group G ( N +2) (0) ⊂ G 0 of all diﬀeomorphisms whose ( N + 2)- jet at 0 has th e form J N +2 0 ( ϕ ) i ( x ) = x i + 1 ( N +2)! P j,k ,α 1 ,...,α N +2 ξ i j kα 1 ...α N x j x k x α 1 · · · x α N , ξ i j kα 1 ...α N ∈ C n N +3 , ξ i j kα 1 ...α N = ξ i k j α σ (1) ...α σ ( N ) , ∀ p ermutat ion σ , and u sing (1.27) instead of (1.26), one deriv es as ab ov e the v anishing of all co eﬃcien ts c I ,κ with | κ | = N . This lo wers the h eigh t in (1.34) and thus completes the induction. Let B d enote th e ideal of A ( h n ) generate d by the com binations of the form δ i j ℓ k − δ i j k ℓ − δ s j k δ i sℓ + δ s j ℓ δ i sk , (1.35) whic h according to (1.28) v anish when viewed in H n . 14 Corollary 1.4. The algebr a H n is isomorp hic to the quotient of the univer- sal enveloping algebr a A ( h n ) b y the ide al B . Pr o of. Extending the notat ion for indices introd uced ab o v e, we form multi- indices of the third kind, K ′ = { κ ′ 1 ≤ . . . ≤ κ ′ r } , which are similar to th ose of the second kind (1.30) except that w e drop the r equiremen t k s ≤ ℓ s 1 , i.e. κ ′ s =  i s j s ≤ k s , ℓ s 1 ≤ . . . ≤ ℓ s p s  , s = 1 , . . . , r . (1.36) W e then form a P oincar ´ e-Birkhoﬀ-Witt basis of A ( h n ) out of the monomials δ K ′ Z I , ord ered lexicographically . Let π : A ( h n ) → H n b e the tautologi - cal algebra homomorphism, wh ic h sends the generators of h n to the same sym b ols in H n . In p articular, it sends the basis elements of A ( h n ) to the corresp ondin g elemen ts in H n π ( δ K ′ Z I ) = δ K ′ Z I , but the monomial in the righ t hand side b elongs to the basis of H n only when the comp onen ts of all κ ′ s ∈ K ′ are in the increasing order. Eviden tly , B ⊂ K er π . T o prov e the con ve rse, assume that u = X I ,K ′ c I ,K ′ δ K ′ Z I ∈ A ( h n ) satisﬁes π ( u ) ≡ X I ,K ′ c I ,K ′ δ K ′ Z I = 0 . (1.37) In ord er to sho w that u b elongs to the ideal B , w e shall again use indu ction, on the h eigh t of u . The height 0 case is ob vious, b ecause the 0-heigh t monomials r emain linea rly indep end en t in H n , so π ( u ) = 0 imp lies c I = 0 for eac h I , and therefore u = 0. Let no w N ≥ 1 b e the largest height of o ccur ring in u . F or eac h K ′ of heigh t N , denote by K the m ulti-index with the corresp ond ing comp onen ts κ ′ s ∈ K ′ rearranged in the increasing order. In vie w of (1.28), one can replace eac h δ K ′ in the equation (1.3 7 ) by δ K + lower height , b ecause the diﬀerence b elongs to B . Th us, th e top heigh t part of u b ecomes X I , | K | = N c I ,K ′ δ K Z I , and so u = v + X I , | K | = N c I ,K ′ δ K Z I (mo d B ) , 15 where v has heigh t at most N − 1. Then (1.37 ) tak es the form π ( v ) + X I , | K | = N c I ,K ′ δ K Z I = 0 , and f rom Pr op osition 1.3 it follo ws that the coeﬃcien t of eac h δ K Z I v an- ishes. One concludes that u = v (mo d B ) . On the other hand, b y the indu ction hyp othesis, π ( v ) = 0 implies v ∈ B . In order to stat e the next resu lt, we associate to an y element h 1 ⊗ . . . ⊗ h p ∈ H ⊗ p n a multi-diﬀer ential op erator, acting on A , by the follo wing form ula T ( h 1 ⊗ . . . ⊗ h p ) ( a 1 ⊗ . . . ⊗ a p ) = h 1 ( a 1 ) · · · h p ( a p ) , (1 .38) where h 1 , . . . , h p ∈ H n and a 1 , . . . , a p ∈ A ; the linear extension of this assignment will b e d en oted by the same letter. Prop osition 1.5. F or e ach p ∈ N , the line ar tr ansformation T : H ⊗ p n − → L ( A ⊗ p , A ) is inje ctive. Pr o of. F or p = 1, T giv es the standard action of H n on A , whic h w as ju st sho wn to b e faithful. T o pr o v e that K er T = 0 for an arbitrary p ∈ N , assume that H = X ρ h 1 ρ ⊗ · · · ⊗ h p ρ ∈ Ker T . After ﬁxing a P oincar ´ e-Birkhoﬀ-Witt basis as ab ov e, w e may uniqu ely ex- press eac h h j ρ in the form h j ρ = X I j ,K j C ρ, I j ,K j δ K j Z I j , with C ρ, I j ,K j ∈ C . Ev aluat ing T ( H ) o n elemen tary tensors of the form f 1 U ∗ ϕ 1 ⊗ · · · ⊗ f p U ∗ ϕ p , one obtains X ρ,I ,K C ρ, I 1 ,K 1 · · · C ρ, I p ,K p δ K 1  Z I 1 ( f 1 ) U ∗ ϕ 1  · · · δ K p  Z I p ( f p ) U ∗ ϕ p  = 0 . Ev aluat ing further at a p oint u 1 = ( x 1 , y 1 ) ∈ F R n , and denoting u 2 = e ϕ 1 ( u 1 ) , . . . , u p = e ϕ p − 1 ( u p − 1 ) , 16 the ab ov e iden tit y giv es X ρ,I ,K C ρ, I 1 ,K 1 · · · C ρ, I p ,K p · γ K 1 ( ϕ 1 )( u 1 ) · · · γ K p ( ϕ p )( u p ) · Z I 1 ( f 1 )( u 1 ) · · · Z I p ( f p )( u p ) = 0 . Let us ﬁx p oin ts u 1 , . . . , u p ∈ F R n and then diﬀeomorphisms ψ 0 , ψ 1 , . . . , ψ p , suc h that u 2 = e ψ 1 ( u 1 ) , . . . , u p = e ψ p − 1 ( u p − 1 ) . F ollo wing a line of reasoning similar to that of the preceding pro of, and iterated with r esp ect to the p oin ts u 1 , . . . , u p , we can infer that for eac h p -tuple of ind ices of the ﬁrst kind ( I 1 , . . . , I p ) one h as X ρ,K C ρ, I 1 ,K 1 · · · C ρ, I p ,K p · γ K 1 ( ϕ 1 )( u 1 ) · · · γ K p ( ϕ p )( u p ) = 0 . Similarly , making rep eated use of diﬀeomorphisms of the form ψ k ◦ ϕ with ϕ ∈ G ( N ) ( u k ) , k = 1 , . . . , p , for su ﬃcien tly man y v alues of N , w e ca n ev en tually conclude that for any ( K 1 , . . . , K p ) X ρ C ρ, I 1 ,K 1 · · · C ρ, I p ,K p = 0 . This pr o v es that H = 0. The crossed pro duct algebra A = C ∞ c ( F R n ) ⋊ G carries a canonical trace, uniquely determined up to a scaling factor. I t is deﬁ n ed as th e linear func- tional τ : A → C , τ ( f U ∗ ϕ ) =        Z F R n f  , if ϕ = I d , 0 , otherwise . (1.39) Here  i s th e vo lume form attac hed to th e canonical framing giv en b y the ﬂat connection  = n ^ k =1 θ k ∧ ^ ( i,j ) ω i j (ordered lexicographically) . 17 The tracial p r op erty τ ( a b ) = τ ( b a ) , ∀ a, b ∈ A , is a consequence of the G -in v ariance of the volume form  . In turn, the latter follo ws from the fact th at e ϕ ∗ ( θ ) = θ and e ϕ ∗ ( ω ) = ω + γ · θ ; indeed, e ϕ ∗ (  ) = n ^ k =1 θ k ∧ ^ ( i,j )  ω i j + γ i j ℓ ( ϕ ) θ ℓ  = n ^ k =1 θ k ∧ ^ ( i,j ) ω i j . This trace s atisﬁes an inv ariance prop erty relativ e to th e mo dular char acter of H n . The latter, δ : H n → C , extends th e tr ac e c haracter of g l ( n, R ), and is deﬁn ed on the algebra generators as follo ws: δ ( Y j i ) = δ j i , δ ( X k ) = 0 , δ ( δ i j k ) = 0 , i, j, k = 1 , . . . , n . (1.4 0) Clearly , this deﬁnition is compatible with the relations (1.28) and therefore extends to a c haracter of th e algebra H n . Prop osition 1.6. F or any a, b ∈ A and h ∈ H n one has τ ( h ( a )) = δ ( h ) τ ( a ) . (1.41) Pr o of. It suﬃces to v erify the state d identit y on the alge br a generators of H n . Evid en tly , b oth sides v anish if h = δ i j k . O n the other hand, its restric- tion to the Lie alge br a g = R n ⋊ gl ( n, R ) is just the restatemen t, at the lev el of the Lie al gebra, of the in v ariance prop ert y of the left Haa r m easur e on G = R n ⋊ GL( n, R ) with resp ect to righ t translations. Prop osition 1.7. Ther e exists a unique anti-automo rphism e S : H n → H n such that τ ( h ( a ) b ) = τ ( a e S ( h )( b )) , (1.42) for any h ∈ H n and a, b ∈ A . Mor e over, e S is involutive: e S 2 = Id . (1.43) 18 Pr o of. Using the ‘Leibnitz rule’ (1.12) for vertic al ve ctor ﬁelds, and the in v ariance prop ert y (1.41) applied to the pro du ct ab , a, b ∈ A , one obtains τ ( Y j i ( a ) b ) = − τ ( a Y j i ( b )) + δ j i τ ( a b ) , ∀ a, b ∈ A . (1.44) On the other hand, for the basic horizon tal vec tor ﬁelds, (1.17) and (1 .41) giv e τ ( X k ( a ) b ) = − τ ( a X k ( b )) − τ ( δ i j k ( a ) Y j i ( b )) = − τ ( a X k ( b )) + τ ( a δ i j k ( Y j i ( b )) ; (1.45) the second equalit y u ses the 1-co cycle nature of γ i j k . The same p rop erty implies τ ( δ i j k ( a ) b ) = − τ ( a δ i j k ( b )) , ∀ a, b ∈ A . (1.46) Th us, the generators of H n satisfy inte gr ation by p arts iden tities of the f orm (1.42), with e S ( Y j i ) = − Y j i + δ j i (1.47) e S ( X k ) = − X k + δ i j k Y j i (1.48) e S ( δ i j k ) = − δ i j k (1.49) Since the pairing ( a, b ) 7→ τ ( a b ) is non-degenerate, the ab o v e op erators are uniquely determined . Being obviously multiplic ativ e, the ‘in tegration b y p arts’ rule extend s from generators to all elemen ts h ∈ H n , and uniquely d eﬁnes a map e S : H n → H n satisfying (1.42). In tur n, this very identit y imp lies t hat e S is a h omomor- phism fr om H n to H op n , as wel l as the fact th at e S is in v olutiv e. Relying on the ab o ve results, we are n o w in a p osition to equip H n with a canonical Hopf structure. Theorem 1.8. Ther e exists a u nique Hopf algebr a structur e on H n with r e sp e ct to which A is a left H n -mo dule a lgebr a. Pr o of. The formulae (1.12), (1.17) and (1.19 ) extend b y multiplic ativ ely to a general ‘Leibnitz rule’ satisﬁed by an y elemen t h ∈ H n , of the form h ( ab ) = X ( h ) h (1) ( a ) h (2) ( b ) , h (1) , h (2) ∈ H n , a, b ∈ A (1.50) 19 By Prop osition 1.5, this p rop erty uniquely determines th e c opr o duct map ∆ : H → H ⊗ H , ∆( h ) = X ( h ) h (1) ⊗ h (2) , (1.51) that satisﬁes T (∆ h )( a ⊗ b ) = h ( ab ) . (1.52) F ur th ermore, the coasso ciativit y of ∆ b ecomes a consequence o f the asso- ciativit y of A , b ecause after applying T it amoun ts to the ident it y h (( ab ) c ) = h ( a ( bc )) , ∀ h ∈ H n , a, b ∈ A . Similarly , the prop erty that ∆ is an algebra homomorphism follo ws from th e fact th at A is a left H n -mo dule. By the very d eﬁ nition of the copro duct, A is actually a left H n - mo dule algebr a . The c ounit is d eﬁ ned b y ε ( h ) = h (1) ; (1.53) when transp orted via T , its requir ed prop erties amoun t to the ob vious iden- tities h ( a 1) = h (1 a ) = h ( a ) , ∀ h ∈ H n , a ∈ A . It remains t o sho w the existence of antip o de . W e ﬁrst c hec k that the anti- automorphism e S is a twiste d antip o de , i.e. satisﬁes for an y h ∈ H , (Id ∗ e S )( h ) := X ( h ) h (1) e S ( h (2) ) = δ ( h ) 1 , (1.54) ( e S ∗ Id)( h ) := X ( h ) e S ( h (1) ) h (2) = δ ( h ) 1 . (1.55) Indeed, with a, b ∈ A arbitrary , one has τ ( a δ ( h ) b )) = τ ( h ( ab )) = X ( h ) τ ( h (1) ( a ) h (2) ( b )) = X ( h ) τ ( a ( e S ( h (1) ) h (2) )( b )) , whic h pr o v es (1.54) . Similarly , but also usin g the tracial prop ert y , τ ( a δ ( h ) b )) = τ ( h ( ba )) = X ( h ) τ ( h (2) ( a ) h (1) ( b )) = X ( h ) τ ( a ( e S ( h (2) ) h (1) )( b )) , or equiv alen tly X ( h ) e S ( h (2) ) h (1) = δ ( h ) 1; 20 applying e S to b oth sides yields (1.55) . No w let ˇ δ ∈ H ∗ n denote th e con v olution inv erse of the c haracter δ ∈ H ∗ n , whic h on generators is give n b y ˇ δ ( Y j i ) = − δ j i , ˇ δ ( X k ) = 0 , ˇ δ ( δ i j k ) = 0 , i, j, k = 1 , . . . , n . (1.56) Then S := ˇ δ ∗ e S is an algebra an ti-homorphism whic h satisﬁes the an tip o d e requirement X ( h ) S ( h (1) ) h (2) = ε ( h ) 1 = X ( h ) h (1) S ( h (2) ) on the generators, and hence for any h ∈ H n . 1.2 The general case of a ﬂat primitiv e pseudogroup Let Π b e a ﬂat pr im itive Lie pseudogroup of lo cal C ∞ -diﬀeomorphisms of R m . Denote by F Π R m the sub-b undle of F R m consisting of the Π-frames on R m . It consists of th e 1-jets at 0 ∈ R n of the germs of lo cal diﬀeomorphisms φ ∈ Π. Since Π cont ains the translatio ns, F Π R m can be iden tiﬁed, b y the restriction of the map ( 1.1 ), to R m × G 0 (Π), where G 0 (Π) ⊂ GL( m, R ) is the linea r isotrop y group, formed of the J acobians at 0 of the local diﬀeo- morphisms φ ∈ Π preserving the origin. The ﬂat c onnection on F R m restricts to a connection form on F Π R m with v al ues in the Lie algebra g 0 (Π) of G 0 (Π), ω Π := y − 1 d y ∈ g 0 (Π) , y ∈ G 0 (Π) . The basic horizont al v ector ﬁ elds on F Π R m are restrictions of those on F R m , X k = y µ k ∂ ∂ x µ , k = 1 , . . . , 2 n , y = ( y j i ) ∈ G 0 (Π) , (1.57) and the f undamental vertica l v ector ﬁ elds are Y j i = y µ i ∂ ∂ y µ j , i, j = 1 , . . . , 2 n, y = ( y j i ) ∈ G 0 (Π); ( 1.58) when assem bled in to a matrix-v alued vecto r ﬁeld, Y =  Y j i  tak es v a lues in the Lie s ubalgebra g 0 (Π) ⊂ gl ( m, R ). By virtue of Prop osition 1.1, (1.57), (1.58) are also left-in v aria nt v ector ﬁelds, th at give a framing of the group of aﬃn e Π-motions G (Π) := R m ⋊ G 0 (Π). 21 The group G (Π) := Diﬀ ( R m ) ∩ Π of glo bal Π-diﬀeomorphisms acts on F Π R m b y prolongation, and the corresp onding crossed pro d uct algebra A (Π) := C ∞ c ( F Π R m ) ⋊ G (Π) is a subalgebra of A . After promoting the ab o ve v ector ﬁelds X k and Y j i to linear trans formations in L ( A (Π)), one automatically obtains, as in § 1.1, the aﬃliated m ultiplication op erators δ i j k ∈ L ( A (Π)), δ i j k ( f U ∗ ϕ ) = γ i j k ( ϕ ) f U ∗ ϕ , ϕ ∈ G (Π) , and then their higher ‘deriv ativ es’ δ i j k ℓ 1 ...ℓ r ∈ L ( A (Π)), δ i j k ℓ 1 ...ℓ r ( f U ∗ ϕ ) = γ i j k ℓ 1 ...ℓ r ( ϕ ) f U ∗ ϕ , ∀ ϕ ∈ G (Π) . These are precisely the restrictions of the corresp onding op er ators in L ( A ), c haracterized b y the pr op ert y t hat the m × m -matrix d eﬁned by their ‘isot ropy part’ is g 0 (Π)-v alued:  γ i j •···• ( ϕ )  1 ≤ i,j ≤ m ∈ g 0 (Π) ⊂ gl ( m, R ) , ϕ ∈ G (Π) . (1.5 9) They form a Lie sub algebra h (Π) of h m and satisfy the Bianc hi-type iden ti- ties of Pr op osition 1.2. W e let H (Π) denote th e generated subalgebra of L ( A (Π)) generated by the ab o v e op er ators, while B (Π) stands f or the ideal generated by the iden tities (1.28). Theorem 1.9. The algebr a H (Π) is isomorphic to the quotient of the uni- versal enve loping algebr a A  h (Π)  by the ide al B (Π) , and c an b e e quipp e d with a uni q ue Hopf algebr a structur e with r esp e ct to which A (Π) is a left H (Π) - mo dule algebr a. Pr o of. The pro of amounts to a mere rep etition of the steps follo we d in the previous su bsection to establish Theorem 1.8. It suﬃ ces to notice that all the argum en ts remain v alid when the isotropy-t yp e o f the generato rs is restricted to the linear isotrop y Lie algebra g 0 (Π). T o illustrate the construction of the Hopf algebra H (Π) in a concrete f ash - ion, we close this section with a more detailed discussion of the t w o main sub classes of ﬂat primitiv e Lie pseudogroups: vo lume preserving and sym- plectic. 22 1.2.1 Hopf algebra of the volume pre serving pseudogroup The su b -bund le F s R n of F R n consists in this case of all sp e cial (unimo du lar) frames on R n , deﬁned b y taking the 1-jet at 0 ∈ R n of germs of lo cal diﬀeomorphisms φ on R n that preserve the vo lume form, i.e. φ ∗ ( dx 1 ∧ · · · ∧ dx n ) = dx 1 ∧ · · · ∧ dx n . By means of the iden tiﬁcation (1.1 ), F s R n ≃ R n × SL( n, R ). The ﬂat conn ection is giv en by the sl ( n, R )-v alued 1-form s ω = ( s ω i j ) s ω i j := ω i j = ( y − 1 ) i µ d y µ j , i 6 = j = 1 , . . . , n ; s ω i i := ω i i − ω n n , i = 1 , . . . , n − 1 . (1.60) The basic horizon tal v ector ﬁelds on F s R n are restrictio ns of th ose on F R n , s X k = y µ k ∂ µ , k = 1 , . . . , n , y ∈ SL( n, R ) while the fu ndamenta l vertica l v ector ﬁelds are s Y j i := Y j i = y µ i ∂ j µ , i 6 = j = 1 , . . . , n, s Y i i := 1 2  Y i i − Y n n  , i = 1 , . . . , n − 1 . By Prop osition 1.1, these are also the left-inv arian t ve ctor ﬁelds on the group G s := R n ⋊ S L( n, R ) associated to th e standard basis of g s := R n ⋊ sl ( n, R ). The group G s := Diﬀ ( R n , v ol) of vo lume preserving d iﬀeomorphisms acts on F s R n as in (1.6), an d the corresp ondin g crossed pro d uct alge br a A s := C ∞ c ( F s R n ) ⋊ G s is a sub algebra of A . The analogue of (1.14) is U ∗ ϕ s X k U ϕ = s X k − s γ i j k ( ϕ ) s Y j i , (1.61) where s γ i j k = γ i j k , i 6 = j = 1 , . . . , n, and s γ i ik = γ i ik − γ n nk , i = 1 , . . . , n − 1 . One thus obtains the m ultiplication op erators s δ i j k ℓ 1 ...ℓ r ∈ L ( A s ), s δ i j k ℓ 1 ...ℓ r ( f U ∗ ϕ ) = s γ i j k ℓ 1 ...ℓ r ( ϕ ) f U ∗ ϕ , 23 whic h conti nue to satisfy the Bianc hi-t yp e identitie s of Prop osition 1.2. Denoting b y S H n the subalgebra of L ( A s ) ge nerated b y the abov e o p erators, one equips it with the canonical Hopf structure with resp ect to w h ic h A s is a left S H n -mo dule algebra. W e remark that the Hopf algebra S H n is unimo dular , in the sense that its an tip o de is inv olutiv e. The c onforma l volume pr eserving case is similar, except that the linear isotrop y subgroup is CSL( n, R ) = R + × SL( n, R ). 1.2.2 Hopf algebra of the symplectic pseudogroup Let G sp ⊂ G b e the subgroup of diﬀeomorphisms of R 2 n preserving th e symplecting form, Ω = dx 1 ∧ dx n +1 + · · · + dx n ∧ dx 2 n . (1.62) Denote b y F sp R 2 n the sub-bu ndle of F R 2 n formed of symplectic frames on R 2 n , i.e. those deﬁn ed by taking th e 1-jet a t 0 ∈ R 2 n of germs of lo cal dif- feomorphisms φ on R 2 n preserving the form Ω. Via (1.1) it can b e iden tiﬁed to R 2 n × Sp( n, R ). In turn, Sp( n, R ) is identiﬁed to the subgroup of matrices A ∈ GL(2 n, R ) satisfying t A J A = J, where J =  0 Id n − Id n 0  , (1.63) while its Lie algebra sp ( n, R ) is formed of matrices a ∈ gl (2 n, R ) su ch that t a J + J a = 0 , (1.64) Th us the ﬂat connection is giv en by the sp ( n, R )-v alued 1-form sp ω := y − 1 d y ∈ sp ( n, R ) , y ∈ Sp( n, R ) , the basic horizo ntal v ecto r ﬁelds on F sp R 2 n are restrictio ns of th ose on F R 2 n , sp X k = y µ k ∂ µ , k = 1 , . . . , 2 n , y ∈ Sp( n, R ) , and the fu ndament al ve rtical v ect or ﬁelds are giv en by th e sp ( n, R )-v alued v ector ﬁ eld ( sp Y j i ) ∈ sp ( n, R ) sp Y j i = y µ i ∂ j µ , i, j = 1 , . . . , 2 n, y ∈ Sp( n, R ) . They also form a basis of left-in v arian t v ector ﬁelds on the group G sp := R 2 n × Sp( n, R ) asso ciated to the standard basis of g sp := R 2 n ⋊ sp ( n, R ). 24 The group G sp := Diﬀ ( R n , Ω) of symplectic diﬀeomorphism s acts on F sp R 2 n b y prolongation, and the corresp onding crossed pro d uct algebra A sp := C ∞ c ( F sp R 2 n ) ⋊ G sp is a subalgebra of A . T he v ector ﬁelds sp X k and sp Y j i extend to linear transformations in L ( A sp ), and their actio n brings in multiplicatio n op era- tors sp δ i j k ℓ 1 ...ℓ r ∈ L ( A sp ), sp δ i j k ℓ 1 ...ℓ r ( f U ∗ ϕ ) = sp γ i j k ℓ 1 ...ℓ r ( ϕ ) f U ∗ ϕ . W e le t S p H n denote the generated sub algebra of L ( A sp ) generated by the ab o v e op erators. I t acquires a un ique Hopf algebra structure such that A sp is a le ft S p H n -mo dule a lgebra. Like S H n , the Hopf algebra S p H n is unimo du lar. The c onformal symple c tic case is again similar, except t hat the linear isotropy subgroup is CS p( n, R ) = R + × Sp( n, R ). 1.3 Hopf algebra of t he contact pseudogroup W e denote b y G cn ⊂ G the subgroup of orientat ion preserving diﬀeomor- phisms of R 2 n +1 whic h lea v e inv ariant the con tact f orm α := − dx 0 + 1 2 n X i =1 ( x i dx n + i − x n + i dx i ) . (1.65) The v ector ﬁ eld E 0 := ∂ 0 ≡ ∂ ∂ x 0 satisﬁes ι E 0 ( α ) = 1 and ι E 0 ( dα ) = 0, i.e. represents the R e eb ve ctor ﬁeld of the con tact structure. Th e c ontact distribution D := Ker α is spanned by the v ector ﬁ elds { E 1 , . . . , E 2 n } , E i = ∂ i − 1 2 x n + i E 0 , E n + i = ∂ n + i + 1 2 x i E 0 , 1 ≤ i ≤ n. (1.66) The Lie brack et s b et wee n these v ector ﬁelds are precisely those of the Heisen- b erg Lie alge br a h n : [ E i , E j + n ] = δ i j E 0 , [ E 0 , E i ] = 0 , [ E 0 , E j + n ] = 0 , i, j = 1 , . . . , n. This giv es an identi ﬁcation of R 2 n +1 , w hose standard basis we denote by { e 0 , e 1 , . . . , e 2 n } , with the Heisen b erg g roup H n , whose group la w is giv en 25 b y x ∗ y = x + y + β ( x ′ , y ′ ) e 0 ≡  x 0 + y 0 + β ( x ′ , y ′ ) , x ′ , y ′  (1.67) where x = ( x 0 , x 1 , . . . , x 2 n ) , y = ( y 0 , y 1 , . . . , y 2 n ) ∈ H n ≡ R 2 n +1 , x ′ = ( x 1 , . . . , x 2 n ) , y ′ = ( y 1 , . . . , y 2 n ) ∈ R 2 n and β ( x ′ , y ′ ) = 1 2 n X i =1 ( x i y n + i − x n + i y i ) . (1.68) Lemma 1.10. The ve ctor ﬁeld s { E 0 , E 1 , . . . , E 2 n } ar e r esp e ctively th e left- invariant ve ctor ﬁelds on H n determine d by the b asis { e 0 , e 1 , . . . , e 2 n } of its Lie algebr a h n . Pr o of. Straigh tforward, giv en that in the ab o v e realization the exp onent ial map e xp : h n → H n coincides with the ident it y map Id : R 2 n +1 → R 2 n +1 . W e recall that a c ontact diﬀe omor phism is a diﬀeomorphism φ : R 2 n +1 → R 2 n +1 suc h that φ ∗ (Ker α ) = Ker α, or equiv al entl y φ ∗ ( α ) = f α for a no where v anishing function f : R 2 n +1 → R ; φ is orien tation pr eserving iﬀ f > 0 and is called a strict c ontact diﬀe omo rphism if f ≡ 1. In particular, the group left translations L a : H n → H n , a ∈ H n , L a ( x ) = a ∗ x =  a 0 + x 0 + β ( a ′ , x ′ ) , a ′ + x ′  ∀ x ∈ H n , are easily seen to b e strict con tact diﬀeomorphism. They r eplace the usual translations in R 2 n +1 . T ogether with the linear transformations p reserving the symp lectic form β : R 2 n × R 2 n → R 2 n of (1.68), S A ( x ) = ( x 0 , A x ′ ) , A ∈ S p( n, R ) , x ′ = ( x 1 , . . . , x 2 n ) , they form a group of strict conta ct transformations whic h is isomorphic to H n ⋊ Sp( n, R ). In deed, one has ( S A ◦ L a )( x ) = S A  x 0 + a 0 + β ( a ′ , x ′ ) , x ′ + a ′  =  x 0 + a 0 + β ( a ′ , x ′ ) , A ( x ′ + a ′ )  =  x 0 + a 0 + β ( A a ′ , A x ′ ) , A ( x ′ + a ′ )  = L S A ( a )  S A ( x )  = ( L S A ( a ) ◦ S A )( x ) , therefore S A ◦ L a ◦ S − 1 A = L S A ( a ) ; also ( L a ◦ S A ) ◦ ( L b ◦ S B ) = L a ◦ ( S A ◦ L b ) ◦ S B = L a ◦ ( L S A ( b ) ◦ S A ) ◦ S B = L a ∗ S A ( b ) ◦ S AB . 26 W e shall enlarge this g roup the 1-parameter group of c ontact homotheties { µ t ; t ∈ R ∗ + } , deﬁned b y µ t ( x ) = ( t 2 x 0 , t x ′ ) , x ∈ H n ; one has µ t ◦ S A = S A ◦ µ t and µ t ◦ L a = L µ t ( a ) ◦ µ t . W e denote by G cn t the group generated by the conta ct transformations { L a , S A , µ t ; a ∈ H n , A ∈ Sp( n, R ) , t ∈ R ∗ + } . It can b e identi ﬁed with the semidirect pro duct G cn ∼ = H n ⋊ CSp( n, R ) , where C Sp( n, R ) = S p( n, R ) × R ∗ + is the c onform al symple c tic gr oup . Via the identiﬁcat ion L a ◦ S A ◦ µ t ∼ = ( a , A, t ) , a ∈ H n , A ∈ S p( n, R ) , t ∈ R ∗ + , (1.6 9) the multiplica tion la w is ( a , A, t ) · ( b , B , s ) = ( a ∗ S A ( µ t ( b )) , AB , ts ) . ( 1.70) Besides the usual tangen t b u ndle T R 2 n +1 , it w ill b e co nv enien t to introd uce a version of it that arises naturally when the conta ct stru cture is treated as sp ecial case of a Heisenb er g m anifold ( cf. e.g. [25]). Denoting by R := T R 2 n +1 / D the line bundle determined b y the class of th e Reeb v ector ﬁ eld mo dulo the conta ct distrib ution, the H-tangent bund le is the direct sum T H R 2 n +1 := R ⊕ D . If φ ∈ G cn is a cont act d iﬀeomorphism, then its tangen t map at x ∈ R 2 n +1 , φ ∗ x : T x R 2 n +1 → T φ ( x ) R 2 n +1 , lea ve s the co ntac t distribution D inv arian t, and hence induces a corresp onding H-tangent map , whic h will b e denoted φ H ∗ x : T H x R 2 n +1 → T H φ ( x ) R 2 n +1 . Lemma 1.11. Given any c on tact diﬀe omor phism φ ∈ G cn , ther e is a unique ϕ ∈ G cn , such that ψ = ϕ − 1 ◦ φ has the pr op erties ψ (0) = 0 and ψ H ∗ 0 = Id : T H 0 R 2 n +1 → T H 0 R 2 n +1 . (1.71) Pr o of. Consider th e H-tangen t map φ H ∗ 0 : T H 0 R 2 n +1 → T H a R 2 n +1 , wher e a = φ (0). T hen r elativ e to the mo ving frame { E 1 , . . . , E 2 n } for D , φ H ∗ 0 | D : 27 D 0 → D a is given by a conform al symplectic matrix A φ (0) ∈ CS p( n, R ) with conformal factor t φ (0) 2 > 0. F urtherm ore, one can easily see that φ ∗ 0 ( e 0 ) = t φ (0) 2 E 0 | a + w φ , with w φ ∈ D a , and so t he restrictio n φ H ∗ 0 | R : R 0 → R a can b e id entiﬁed with the scala r t φ (0) 2 relativ e to the mo ving fr ame E 0 of R . Since A = t − 1 φ (0) A φ (0) ∈ Sp( n, R ), w e can form ϕ := L a ◦ S A ◦ µ t ∈ G cn , a = φ (0) , t = t φ (0) , ( 1.72) and w e claim that this ϕ ∈ G cn fulﬁlls the required prop erty . Ind eed, L − 1 a ◦ ϕ has the expr ession ( L − 1 a ◦ ϕ )( x ) =  t φ (0) 2 x 0 , t φ (0) A x ′  and th erefore the matrix of ϕ ∗ 0 : T 0 R 2 n +1 → T a R 2 n +1 is the same as that of φ H ∗ 0 : T H 0 R 2 n +1 → T H a R 2 n +1 , i.e. ( L − 1 a ◦ ϕ ) ∗ 0 =        t φ (0) 2 0 . . . 0 0 . . . t φ (0) A 0        =        t φ (0) 2 0 . . . 0 0 . . . A φ (0) 0        . (1.73) Th us, ψ = ϕ − 1 ◦ φ satisﬁes (1.71). T o pro v e un iqu eness, observe th at if ϕ = L a ◦ S A ◦ µ t ∈ G cn fulﬁlls (1.71), then a = 0 and, since ϕ ∗ 0 : T 0 R 2 n +1 → T 0 R 2 n +1 is of the form (1.7 3 ), A = Id and t = 1. The su b group of G cn deﬁned by the t wo conditions in (1.71) will b e d en oted N cn . L emma 1.11 giv es a Kac de c omp osition for the group of orien tation preserving con tact diﬀeomorphisms: G cn = G cn · N cn . (1.74) Note though that the group N cn is no longer pro-unip oten t. Ho w ev er, it has a pr o-unip otent n orm al subgrou p of ﬁ nite co dimension, namely U cn := { ψ ∈ N cn | ψ (0) = 0 , ψ ∗ 0 = Id : T 0 R 2 n +1 → T 0 R 2 n +1 } ; this give s rise to a group extension Id → U cn → N cn → R 2 n → 0 , (1.75) 28 with the last arro w give n b y the tangen t map at 0. The deco mp osition (1.74 ) giv es r ise to a pair of acti ons of G cn : a left action ⊲ on G cn ∼ = G cn /N cn , and a righ t action ⊳ on N cn ∼ = G cn \ G cn . T o understand the left action, let φ ∈ G cn and ϕ = L x ◦ S A ◦ µ t ∈ G cn . By Lemm a 1.11, φ ◦ ϕ = ( φ ⊲ ϕ ) ◦ ψ , with φ ⊲ ϕ ∈ G cn and ψ ∈ N cn . W rite φ ⊲ ϕ = L b ◦ S B ◦ µ s . If in (1.72) one rep laces φ b y φ ◦ ϕ then, A φ ◦ ϕ (0) = A φ ( x ) A ϕ (0) = tA φ ( x ) A, t φ ◦ ϕ (0) = t φ ( x ) t, hence φ ⊲ ϕ = L φ ( x ) ◦ S t φ ( x ) − 1 A φ ( x ) A ◦ µ t φ t Using the parametrization (1.69), th e explicit description of the action can b e recorded as follo ws. Lemma 1.12 . L et φ ∈ G cn and ( a , A, t ) ∈ G cn . Then ˜ φ ( x , A, t ) := φ ⊲ ( x , A, t ) = ( φ ( x ) , t − 1 φ ( x ) A φ ( x ) A, t φ ( x ) t ) . (1.76) With the ab o ve notational con ve ntio n, let R ( b ,B ,s ) denote the righ t transla- tion by an ele ment ( b , B , s ) ∈ Sp( n, R ), and let φ ∈ G cn . W e w an t t o un- derstand the comm utation relationship b et w een these t w o transformations. By (1.70 ) and (1.76), one has ( ˜ φ ◦ R ( b ,B ,s ) )( x , A, t ) = ˜ φ  ( x , A, t ) · ( b , B , s )  = ˜ φ ( x ∗ S A ( µ t ( b )) , AB , ts ) =  φ ( x ∗ S A ( µ t ( b ))) , t − 1 φ ( x ) A φ ( x ) AB , t φ ( x ) ts  . On the other hand, ( R ( b ,B ,s ) ◦ ˜ φ )( x , A, t ) = ( φ ( x ) , t − 1 φ ( x ) A φ ( x ) A, t φ ( x ) t ) · ( b , B , s ) = ( φ ( x ) ∗ t − 1 φ ( x ) S A φ ( x ) A ( µ t φ ( x ) t ( b )) , t − 1 φ ( x ) A φ ( x ) AB , t φ ( x ) ts ) . Although these t w o answ ers are in general diﬀeren t, when b = 0 they do coincide, and w e record this f act in th e f ollo wing statemen t. Lemma 1.13. The left action of G cn on G cn c om mutes with the right tr anslations by the elements of the sub gr oup CSp( n, R ) . 29 As in th e ﬂat case, we p ro ceed to asso ciate to the pseudogroup G cn of orien- tation p reserving diﬀeomorphisms of R 2 n +1 a Hopf algebra H (Π cn ), realized via its Hopf act ion on the crossed pro du ct alge br a A (Π cn ) = C ∞ ( G cn ) ⋊ G cn . This type of constru ction actually applies w h enev er one has a Kac d ecom- p osition of th e form (1.11). One starts with a ﬁ xed basis { X i } 1 ≤ i ≤ m for the Lie algebra g cn of G cn . Eac h X ∈ g cn giv es rise to a left-in v arian t v ector ﬁeld X on G cn , whic h is then extended to a linear op erator on A (Π cn ), X ( f U ∗ φ ) = X ( f ) U ∗ φ , φ ∈ G cn . One has U ∗ φ X i U φ = m X j =1 Γ j i ( φ ) X j , i = 1 , . . . , m, (1.77) with Γ j i ( φ ) ∈ C ∞ ( G cn ). The matrix of fu nctions Γ ( φ ) =  Γ j i ( φ )  1 ≤ i,j ≤ m automatica lly sat isﬁes the co cycle identit y Γ ( φ ◦ ψ ) = ( Γ ( φ ) ◦ ψ ) · Γ ( ψ ) , φ, ψ ∈ G cn . (1.78) W e next denote by ∆ j i ( φ ) the follo wing m ultiplication op erator on A (Π cn ): ∆ j i ( f U ∗ φ ) = ( Γ ( φ ) − 1 ) j i f U ∗ φ , i, j = 1 , . . . , m . With this notation, we deﬁne H (Π cn ) as the su balgebra of linear op erators on A (Π cn ) generated b y th e o p erators X k ’s and ∆ j i ’s, i, j, k = 1 , . . . , m . I n particular, H Π con tains all iterated comm utators ∆ j i,k 1 ...k r := [ X k r , . . . , [ X k 1 , ∆ j i ] . . . ] , i.e. the m ultiplication op erators b y the fu n ctions on G , Γ j i,k 1 ...k r ( φ ) := X k r . . . X k 1 (Γ j i ( φ )) , φ ∈ G . Lemma 1.14 . F or any a, b ∈ A (Π cn ) , one has X k ( ab ) = X k ( a ) b + X j ∆ j k ( a ) X j ( b ) , (1.79) ∆ j i ( ab ) = X k ∆ k i ( a ) ∆ j k ( b ) . (1.80) 30 Pr o of. With a = f 1 U ∗ φ 1 , b = f 2 U ∗ φ 2 , and assembling th e X k ’s in to a column v ector X and the ∆ j i ’s into a matrix ∆ , one has X ( a · b ) = X ( f 1 U ∗ φ 1 f 2 U ∗ φ 2 ) = X ( f 1 U ∗ φ 1 f 2 U φ 1 ) U ∗ φ 2 φ 1 = = X ( f 1 ) U ∗ φ 1 f 2 U ∗ φ 2 + f 1 X ( U ∗ φ 1 f 2 U φ 1 ) U ∗ φ 1 U ∗ φ 2 = = X ( a ) b + f 1 U ∗ φ 1 ( U φ 1 X U ∗ φ 1 )( f 2 ) U ∗ φ 2 = [us in g (1.77)] = X ( a ) b + f 1 U ∗ φ 1 Γ ( φ − 1 1 ) X ( f 2 ) U ∗ φ 2 = = X ( a ) b + f 1 ( Γ ( φ − 1 1 ) ◦ φ 1 ) U ∗ φ 1 X ( f 2 ) U ∗ φ 2 = [using (1.78) ] = X ( a ) b + f 1 Γ ( φ 1 ) − 1 U ∗ φ 1 X ( b ) = X ( a ) b + ∆ ( a ) X ( b ) , whic h pr o v es (1.79) . The id en tit y (1.80) is merely a reformula tion of the co cycle identi t y (1 .78 ). As a consequence, b y m ultiplicativit y ev ery h ∈ H (Π cn ) sa tisﬁes a L eibniz rule of the form h ( ab ) = X h (1) ( a ) h (2) b ) , ∀ a, b ∈ A (Π) . (1.81) Prop osition 1.15. The op er ators ∆ • •···• satisfy the (Bianchi) identities ∆ k i,j − ∆ k j,i = X r,s c k r s ∆ r i ∆ s j − X ℓ c ℓ ij ∆ k ℓ , (1.82) wher e c i j k ar e th e structur e c onstants of g cn , [ X j , X k ] = X i c i j k X i . (1.83) Pr o of. Applying (1.79) one h as, for an y a, b ∈ A (Π cn ), X i X j ( a b ) = X i  X j ( a ) b + X s ∆ s j ( a ) X s ( b )  = = X i ( X j ( a )) b + X r ∆ r i ( X j ( a )) X r ( b ) + X s X i (∆ s j ( a )) X s ( b ) + X r,s (∆ r i (∆ s j ( a )) X r ( X s ( b )) , 31 and thus the comm utators can b e expressed as f ollo ws: [ X i , X j ]( a b ) = [ X i , X j ]( a ) b + X r ∆ r i ( X j ( a )) X r ( b ) − X s ∆ s j ( X i ( a )) X s ( b ) + X s X i (∆ s j ( a )) X s ( b ) − X r X j (∆ r i ( a )) X r ( b ) + + X r,s (∆ r i (∆ s j ( a )) X r ( X s ( b )) − X r,s (∆ r j (∆ s i ( a )) X r ( X s ( b )) = [ X i , X j ]( a ) b − X r  ∆ r i,j ( a ) − ∆ r j,i ( a )  X r ( b ) + + X r,s (∆ r i ∆ s j )( a ) X k c k r s X k ( b ) . On the other hand, by (1.83), the left hand side equals X ℓ c ℓ ij X ℓ ( a b ) = X ℓ c ℓ ij X ℓ ( a ) b + X ℓ c ℓ ij ∆ k ℓ ( a ) X k ( b ) = [ X i , X j ]( a ) b + X ℓ c ℓ ij ∆ k ℓ ( a ) X k ( b ) . Equating the t w o expressions one obtains after cancelat ion P ℓ c ℓ ij ∆ k ℓ ( a ) X k ( b ) = − P k  ∆ k i,j ( a ) − ∆ k j,i ( a )  X k ( b ) + + P k P r,s c k r s (∆ r i ∆ s j )( a ) X k ( b ) . Since a, b ∈ A (Π) are arbitrary and the X k ’s are linearly ind ep endent, this giv es th e claimed ident it y . Let H cn b e the Lie algebra generated by the op erators X k and ∆ j i,k 1 ...k r , i, j, k 1 . . . k r = 1 , . . . , m , r ∈ N . F ollo wing t he same line of a rguments as in the pr o of of Corollary 1.4, one can establish its exact analog. Prop osition 1.16. The algebr a H (Π cn ) is isomorphic to the quotient of the universal enveloping algebr a U ( H cn ) by the ide al B cn gener at e d by the Bianchi identities (1.82 ) . Actually , one can b e quite a bit more sp eciﬁc about the ab ov e co cycles as w ell as ab out the corresp onding Bianc hi iden tities, if one u ses an appropriate basis of the Lie algebra g cn . Recalling that g cn is a semidirect pro du ct of the Heisenberg Lie algebra h n b y the Lie algebra g csp of the conformal symplectic group CS p( n, R ), one can choose the b asis { X i } 1 ≤ i ≤ m suc h that 32 the ﬁr st 2 n + 1 vec tors are the basis { E i } 0 ≤ i ≤ 2 n of h n , wh ile the r est form the canonical basis { Y j i , Z } of g csp , with Z ce ntral. By Lemma 1.13, for an y φ ∈ G cn , U ∗ φ Y U φ = Y , Y ∈ g csp . (1.84) Th us, the element s of g csp act as deriv a tions on A (Π), an d therefore giv e rise to ‘tensorial id en tities’. The only gen uine ‘Bianc hi ident ities’ among (1.82) are those g enerated b y the lifts of th e canonical framing { E 0 , E 1 , . . . , E 2 n } of T H n to left-inv arian t v ector ﬁelds { X 0 , X 1 , . . . , X 2 n } on G cn . Prop osition 1.17. The left- invariant ve ctor ﬁe lds on G cn c or r esp onding to the c ano nic al b asis of the Heisenb er g Lie algebr a ar e as fol lows: X 0 | ( x , A , s ) = s 2 ∂ ∂ x 0 = s 2 E 0 , (1.85) X j | ( x , A , s ) = s 2 n X i =1 a i j E i , 1 ≤ j ≤ 2 n. (1.86) Pr o of. W e start with the lift of E 0 . Since exp( t e 0 ) = t e 0 , o ne has for any F ∈ C ∞ ( G cn ), X 0 F ( x , A, s ) = d dt | t =0 F (( x , A, s ) · ( t e 0 , Id , 1)) = = d dt | t =0 F ( x ∗ tS A ( µ s ( e 0 )) , A, s ); as S A ( µ s ( e 0 )) = s 2 e 0 , w e can con tin ue as follo ws: = d dt | t =0 F ( x ∗ ts 2 e 0 , A, s ) = d dt | t =0 F ( x 0 + ts 2 , x 1 , . . . , x 2 n ) , A, s ) = = s 2 ∂ F ∂ x 0 ( x , A, s ) . This pr o v es (1.85). Next, for 1 ≤ j ≤ 2 n , let X j denote the lift of E j to G cn . Again, usin g that exp( t e j ) = t e j in H n , one has X j F ( x , A, s ) = d dt | t =0 F (( x , A, s ) · ( t e j , Id , 1)) = = d dt | t =0 F ( x ∗ tS A ( µ s ( e j )) , A, s ); 33 b ecause S A ( µ s ( e j )) = s a j , with a j denoting the j th column in the m atrix A , the ab o v e is equal to = d dt | t =0 F ( x ∗ ts a j , A, s ) d dt | t =0 F (( x 0 + tsβ ( x ′ , a j ) , x ′ + ts a j ) , A, s ) = d dt | t =0 F ( x 0 + ts 2 n X i =1 ( x i a n + i j − a i j x n + i ) , x 1 + tsa 1 j , . . . , x 2 n + tsa 1 2 n ) , A, s ) = s 2 n X i =1  x i a n + i j − a i j x n + i  ∂ F ∂ x 0 ( x , A, s ) + s 2 n X k =1 a k ,j ∂ F ∂ x k ( x , A, s ) . Th us, X j | ( x , A , s ) = s 2 n X i =1  x i a n + i j − a i j x n + i  ∂ ∂ x 0 + s 2 n X k =1 a k j ∂ ∂ x k = = s 2 n X i =1  x i a n + i j − a i j x n + i  ∂ ∂ x 0 + s n X i =1  a i j ∂ ∂ x i + a n + i j ∂ ∂ x n + i  = s n X i =1 a i j  ∂ ∂ x i − 1 2 x n + i ∂ ∂ x 0  + s n X i =1 a n + i j  ∂ ∂ x n + i + 1 2 x i ∂ ∂ x 0  = s n X i =1  a i j E i + a n + i j E n + i  , whic h is the expression in (1.86). Remark 1.18. The formulae (1.85) , (1.86) , which taken to ge ther ar e the exact analo gu e of the formula (1.4 ) , simply expr ess the fact that th e tr ansi- tion matrix fr om the b asis { E 0 , E 1 , . . . , E 2 n } to the b a sis { X 0 , X 1 , . . . , X 2 n } of the horizonta l subsp ac e of T ( x ,A,s ) G cn ∼ = T x H n is pr e cisely the matrix        s 2 0 . . . 0 0 . . . sA 0        . W e no w giv e a few examples the co cycles Γ j i ( φ ) ∈ C ∞ ( G cn ), φ ∈ G cn , corresp ondin g to the horizon tal v ector ﬁelds { X 0 , X 1 , . . . , X 2 n } . 34 Starting with X 0 , one has ( U ∗ φ X 0 U φ ) F ( x , A, s ) = ( U ∗ φ s 2 U φ )( U ∗ φ ∂ ∂ x 0 U φ ) F ( x , A, s ) = = t φ ( x ) 2 s 2 ∂ ∂ y 0 F  φ − 1 ( y ) , t φ − 1 ( y ) − 1 A φ − 1 ( y ) B , t φ − 1 ( y ) z  | ( φ ( x ) , t φ ( x ) − 1 A φ ( x ) A, t φ ( x ) s ) = = t φ ( x ) 2 s 2  ∂ x F · ∂ ( φ − 1 ) ∂ y 0 + ∂ A F · ∂ ∂ y 0  t φ − 1 ( y ) − 1 A φ − 1 ( y )  B + + ∂ s F ∂ ∂ y 0  t φ − 1 ( y ) z   | ( φ ( x ) , t φ ( x ) − 1 A φ ( x ) A, t φ ( x ) s ) = = t φ ( x ) 2 s 2  ∂ x F · ∂ ( φ − 1 ) ∂ y 0 − t φ − 1 ( y ) − 2 ∂ ∂ y 0  t φ − 1 ( y )  ∂ A F ·  A φ − 1 ( y ) B  + + t φ − 1 ( y ) − 1 ∂ A F · ∂ ∂ y 0  A φ − 1 ( y )  B + + ∂ s F ∂ ∂ y 0  t φ − 1 ( y )  z  | ( φ ( x ) , t φ ( x ) − 1 A φ ( x ) A, t φ ( x ) s ) . T aking in to account that t φ − 1 ( φ ( x )) t φ ( x ) = 1 , A φ − 1 ( φ ( x )) A φ ( x ) = Id , (1.87) and ∂ ( φ − 1 ) 0 ∂ x 0 ( φ ( x )) = t φ − 1 ( φ ( x )) 2 = t φ ( x ) − 2 , (1.88) one obtains after ev a luation at ( φ ( x ) , t φ ( x ) − 1 A φ ( x ) A, t φ ( x ) s ) ∈ G cn U ∗ φ X 0 U φ = = X 0 + t φ ( x ) 2 s 2 2 n X i =1 ∂ ( φ − 1 ) i ∂ x 0 ( φ ( x )) ∂ ∂ x i − t φ ( x ) 3 s 2 ∂ t φ − 1 ∂ x 0 ( φ ( x )) ∂ A · A (1.89) + t φ ( x ) 2 s 2 ∂ A · ∂ A φ − 1 ∂ x 0 ( φ ( x )) A φ ( x ) A + ∂ t φ − 1 ∂ x 0 ( φ ( x )) t φ ( x ) 3 s 3 ∂ ∂ s . (1.9 0) T o ﬁnd the c o cycles of the form Γ 0 i ’s, with i = 1 , . . . , 2 n , we use (1.66) to replace the p artial deriv ativ es by the horizon tal vec tor ﬁelds, ∂ i = E i + 1 2 x n + i E 0 , ∂ n + i = E n + i − 1 2 x i E 0 , 1 ≤ i ≤ n, (1.91) 35 and rewrite the second term in th e right hand side of (1.90) as follo ws I I term = t φ ( x ) 2 s 2 n X i =1 ∂ ( φ − 1 ) i ∂ x 0 ( φ ( x )) ∂ ∂ x i + 2 n X i = n +1 ∂ ( φ − 1 ) i ∂ x 0 ( φ ( x )) ∂ ∂ x i ! = t φ ( x ) 2 s 2 n X i =1  ∂ ( φ − 1 ) i ∂ x 0 ( φ ( x )) ( E i + + 1 2 x n + i E 0 ) + ∂ ( φ − 1 ) i ∂ x 0 ( φ ( x )) ( E n + i − 1 2 x i E 0 )  = t φ ( x ) 2 β  ∂ ( φ − 1 ) ′ ∂ x 0 ( φ ( x )) , x ′  X 0 + t φ ( x ) 2 s 2 2 n X i =1 ∂ ( φ − 1 ) i ∂ x 0 ( φ ( x )) E i . W e next in ve rt the formula (1.86), cf. Remark 1.18 , E i | ( x , A , s ) = s − 1 2 n X j =1 ˇ a j i X j , 1 ≤ i ≤ 2 n, (1.92) where (ˇ a i j ) = A − 1 , to obtain I I term = t φ ( x ) 2 β  ∂ ( φ − 1 ) ′ ∂ x 0 ( φ ( x )) , x ′  X 0 + t φ ( x ) 2 s 2 n X i,j =1 ∂ ( φ − 1 ) i ∂ x 0 ( φ ( x )) ˇ a j i X j . W e ha v e thus sho wn th at Γ 0 0 ( φ )( x , A, s ) = Id + t φ ( x ) 2 β  ∂ ( φ − 1 ) ′ ∂ x 0 ( φ ( x )) , x ′  , (1.93) Γ i 0 ( φ )( x , A, s ) = t φ ( x ) 2 s 2 n X j =1 ∂ ( φ − 1 ) j ∂ x 0 ( φ ( x )) ˇ a i j , i = 1 , . . . , 2 n. (1. 94) In particular, when restricted to ψ ∈ N cn and ev al uated at the neutral elemen t e = (1 , Id , 1) ∈ G cn , these co cycles tak e the simple form Γ 0 0 ( ψ )(0 , Id , 1) = Id , (1.95) Γ j 0 ( ψ )(0 , Id , 1) = ∂ ( ψ − 1 ) j ∂ x 0 (0) , j = 1 , . . . , 2 n. (1.96) By comparison with the ﬂat case, these co cycles and their deriv ativ es giv e the only n ew typ e of co ordinate fu nctions on the group ψ ∈ N cn , all the rest b eing completely analog ous to the η • • ... • co ordinates of (2.24) . One last ingredient needed for the construction of the Hopf algebra, is pro- vided by the follo wing lemma. 36 Lemma 1.19. The left Haar volume form of the g r oup G cn is invariant under the action ⊲ of G cn . Pr o of. Up to a constan t factor, the left-in v aria nt v olume form of G cn is giv en, in the co ordinates (1.69), b y  cn := α ∧ dα n ∧  Sp ∧ s − 2( n +1) ds s , (1.97) where  Sp is th e left-in v a riant volume form of S p ( N , R ). Usin g the form ula (1.76) expr essing th e action of φ ∈ G cn on G cn , in conjunction with the left in v ariance of  Sp and the fact that φ ∗ ( α ∧ dα n ) = t 2( n +1) φ α ∧ dα n , one immediately sees that ˜ φ ∗ (  cn ) =  cn . As a consequence, w e can d eﬁne a n in v ariant trace τ = τ cn on the crossed pro du ct alg ebra A (Π cn ) = C ∞ ( G cn ) ⋊ G cn b y precisely the same f ormula (1.39). F urther m ore, the follo wing coun terpart of Prop osition 1.6 holds. Prop osition 1.20. The inﬁnitesimal mo dular char acter δ ( X ) = T r(ad X ) , X ∈ g cn , extends uniquely to a char acter δ = δ cn of H (Π cn ) , and the tr ac e τ = τ cn is H (Π cn ) -invariant r elative to this cha r acter, i.e. τ ( h ( a )) = δ ( h ) τ ( a ) , ∀ a, b ∈ A (Π cn ) . (1 .98) Pr o of. On th e th e canonical b asis of g cn , the c haracter δ tak es the v alues δ ( E i ) = 0 , 0 ≤ i ≤ 2 n, δ ( Y j i ) = 0 , and δ ( Z ) = 2 n + 2; indeed, ad( E i )’s are n ilp oten t, Ad( Y j i )’s are u nimo du lar, and [ Z, E 0 ] = 2 E 0 , [ Z , E i ] = E i , ∀ 1 ≤ i ≤ 2 n, [ Z, Y ] = 0 , ∀ Y ∈ g cn . The rest of the pro of is virtually ident ical to that of Pr op. 1.6. Finally , follo wing the same lin e of arguments whic h led to Th eorem 1.8, one obtains the corresp onding analog. Theorem 1.21. Ther e exists a unique Hopf algebr a structur e on H (Π cn ) , such that its tautolo gic al action makes A (Π cn ) a left mo dule algebr a. 37 2 Bicrossed pro duct realization In this secti on w e reconstruct (or rather deconstruct) the Hopf alg ebra af- ﬁliated to a primitiv e L ie pseudogroup as a bicrossed pro d u ct of a matc hed pair of Hopf algebras. In the particular case of H 1 , this h as b een pr o v ed in [15], by direct algebraic calculations that rely on th e d etailed knowledge of its pr esen tation. By con trast, our metho d is completely geometric and for this reason applicable to the en tire class of Lie p s eudogroups admitting a Kac-t yp e decomp osition. W e recall b elo w the most basic notions concerning the bicrossed p ro du ct construction, referring the reader t o M a jid’s m onograph [23] for a detailed exp osition. Let U and F b e tw o Hopf algebras. A linear map H : U → U ⊗ F , H u = u < 0 > ⊗ u < 1 > , deﬁnes a right c o action , and thus e quip s U with a right F -c omo dule c o al gebr a structure, if the follo wing conditions are satisﬁed for any u ∈ U : u < 0 > (1) ⊗ u < 0 > (2) ⊗ u < 1 > = u (1) < 0 > ⊗ u (2) < 0 > ⊗ u (1) < 1 > u (2) < 1 > (2.1) ǫ ( u < 0 > ) u < 1 > = ǫ ( u )1 . (2.2) One can then form a co crossed pr o duct coalgebra F ◮ < U , that has F ⊗ U as un d erlying v ector space and the follo wing coalgebra structure: ∆( f ◮ < u ) = f (1) ◮ < u (1) < 0 > ⊗ f (2) u (1) < 1 > ◮ < u (2) , (2.3) ǫ ( h ◮ < k ) = ǫ ( h ) ǫ ( k ) . (2.4) In a du al fashion, F is called a lef t U mo dule algebr a , if U acts from the left on F via a left action ⊲ : F ⊗ U → F whic h satisﬁes the follo wing condition for any u ∈ U , and f , g ∈ F : u ⊲ 1 = ǫ ( u )1 (2.5) u ⊲ ( f g ) = ( u (1) ⊲ f )( u (2) ⊲ g ) . (2.6) This time we can endo w the underlyin g v ector space F ⊗ U with an algebra structure, to b e denoted by F > ⊳ U , with 1 > ⊳ 1 as its unit and the pro d uct giv en b y ( f > ⊳ u )( g > ⊳ v ) = f u (1) ⊲ g > ⊳ u (2) v (2.7) 38 U and F are said to form a matche d p air of Hopf algebras if they are equipp ed , as ab o ve , w ith an action and a coaction w hic h satisfy the fol- lo wing compatibilit y conditions: f ollo wing conditions for an y u ∈ U , and an y f ∈ F . ǫ ( u ⊲ f ) = ǫ ( u ) ǫ ( f ) , (2.8) ∆( u ⊲ f ) = u (1) < 0 > ⊲ f (1) ⊗ u (1) < 1 > ( u (2) ⊲ f (2) ) , (2.9) H (1) = 1 ⊗ 1 , (2.10) H ( uv ) = u (1) < 0 > v < 0 > ⊗ u (1) < 1 > ( u (2) ⊲ v < 1 > ) , (2.11) u (2) < 0 > ⊗ ( u (1) ⊲ f ) u (2) < 1 > = u (1) < 0 > ⊗ u (1) < 1 > ( u (2) ⊲ f ) . (2.12) One can then form a new Hopf alg ebra F ◮ ⊳ U , cal led the bicr osse d pr o duct of the matc hed pair ( F , U ) ; it has F ◮ < U as un derlying coalgebra, F > ⊳ U as un d erlying algebra and the antip o de is deﬁ n ed b y S ( f ◮ ⊳ u ) = (1 ◮ ⊳ S ( u < 0 > ))( S ( f u < 1 > ) ◮ ⊳ 1) , f ∈ F , u ∈ U . (2.13) 2.1 The ﬂat case As men tioned in the in tro du ction, the matc hed pair of Hopf algebras arises from a matc hed pair of grou p s, via a splitting ` a la G.I. Kac [18]. Prop osition 2.1. L et Π b e a ﬂat primitive Lie pseudo gr oup of inﬁnite typ e, F Π R m the princip al bund le of Π -fr a mes on R m . Ther e is a c anonic al splitting of the gr oup G = Diﬀ ( R m ) ∩ Π , as a c artesian pr o duct G = G · N , with G ≃ F Π R m the gr oup of aﬃne Π -motions of R m , and N = { φ ∈ G ; φ (0) = 0 , φ ′ (0) = Id } . Pr o of. Let φ ∈ G . Since G cont ains the translations, then φ 0 := φ − φ (0) ∈ G , and φ 0 (0) = 0. Moreo v er, the aﬃne diﬀeomorphism ϕ ( x ) := φ ′ 0 (0) · x + φ (0) , ∀ x ∈ R m (2.14) also b elongs to G , a nd h as the same 1-jet at 0 as φ . Therefore, the diﬀeo- morphism ψ ( x ) := ϕ − 1 ( φ ( x )) = φ ′ 0 (0) − 1 ( φ ( x ) − φ ( 0)) , ∀ x ∈ R m (2.15) b elongs to N , and the canonical d ecomp osition is φ = ϕ ◦ ψ, w ith ϕ ∈ G and ψ ∈ N (2.16) 39 giv en b y (2.14) an d (2.1 5 ). The tw o comp on ents are uniqu ely determined, b ecause evidently G ∩ N = { e } . Rev ersing the order in th e ab ov e decomposition one sim ultaneously obtains t w o w ell-deﬁned op erations, of N on G and of G on N : ψ ◦ ϕ = ( ψ ⊲ ϕ ) ◦ ( ψ ⊳ ϕ ) , for ϕ ∈ G and ψ ∈ N (2.17) Prop osition 2.2. The op er ation ⊲ i s a left action of N on G , and ⊳ is a right action of G on N . Both actions le ave the neutr a l element ﬁxe d. Pr o of. Let ψ 1 , ψ 2 ∈ N and ϕ ∈ G . By (2.17), on the one hand ( ψ 1 ◦ ψ 2 ) ◦ ϕ =  ( ψ 1 ◦ ψ 2 ) ⊲ ϕ  ◦  ( ψ 1 ◦ ψ 2 ) ⊳ ϕ  , and on the other hand ψ 1 ◦ ( ψ 2 ◦ ϕ ) = ψ 1 ◦ ( ψ 2 ⊲ ϕ ) ◦ ( ψ 2 ⊳ ϕ ) =  ψ 1 ⊲ ( ψ 2 ⊲ ϕ )  ◦  ψ 1 ⊳ ( ψ 2 ⊲ ϕ )  ◦ ( ψ 2 ⊳ ϕ ) . Equating the resp ectiv e comp on ents in G and N one obtains: ( ψ 1 ◦ ψ 2 ) ⊲ ϕ = ψ 1 ⊲ ( ψ 2 ⊲ ϕ ) ∈ G, resp. (2.18) ( ψ 1 ◦ ψ 2 ) ⊳ ϕ =  ψ 1 ⊳ ( ψ 2 ⊲ ϕ )  ◦ ( ψ 2 ⊳ ϕ ) ∈ N . (2.19) Similarly , ψ ◦ ( ϕ 1 ◦ ϕ 2 ) =  ψ ⊲ ( ϕ 1 ◦ ϕ 2 )  ◦  ψ ⊳ ( ϕ 1 ◦ ϕ 2 )  , while ( ψ ◦ ϕ 1 ) ◦ ϕ 2 = ( ψ ⊲ ϕ 1 ) ◦ ( ψ ⊳ ϕ 1 ) ◦ ϕ 2 = ( ψ ⊲ ϕ 1 ) ◦  ( ψ ⊳ ϕ 1 ) ⊲ ϕ 2  ◦  ( ψ ⊳ ϕ 1 ) ⊳ ϕ 2 )  , whence ψ ⊳ ( ϕ 1 ◦ ϕ 2 ) = ( ψ ⊳ ϕ 1 ) ⊳ ϕ 2 ∈ N , r esp. (2.20) ψ ⊲ ( ϕ 1 ◦ ϕ 2 ) = ( ψ ⊲ ϕ 1 ) ◦  ( ψ ⊳ ϕ 1 ) ⊲ ϕ 2  ∈ G. (2.21) Sp ecializing ϕ = e , resp. ψ = e , in the deﬁn ition (2.17) , one sees that e = Id acts trivially , and at the same time that b oth actions ﬁx e . 40 Via the identiﬁcat ion G ≃ F Π R m , one can recognize the action ⊲ as the usual action of d iﬀeomorphisms on th e frame bun d le, cf. (1.6). Lemma 2.3. The left action ⊲ of N on G c oincides with the r estriction of the natur al action of G on F Π R m . Pr o of. Let φ = ψ ⊲ ϕ ∈ G , with ψ ∈ N and ϕ ∈ G . The asso ciated fr ame, cf. (1.1 ), is  φ (0) , φ ′ (0)  . By (2.17), ψ  ϕ (0)  = ( ψ ⊲ ϕ )  ( ψ ⊳ ϕ )(0)  = φ (0) , since ( ψ ⊳ ϕ )(0) = 0. On diﬀerentiat ing (2.17) at 0 one obtains ψ ′  ϕ (0)  · ϕ ′ (0) = ( ψ ⊲ ϕ ) ′  ( ψ ⊳ ϕ )(0)  · ( ψ ⊳ ϕ ) ′ (0) = ( ψ ⊲ ϕ ) ′ (0) = φ ′ (0) , since ( ψ ⊳ ϕ ) ′ (0) = Id. Th us,  φ (0) , φ ′ (0)  = ˜ ψ  ϕ (0) , ϕ ′ (0)  , as in the deﬁnition (1.6). Deﬁnition 2.4. The co ordin ates of ψ ∈ N ar e the c o eﬃcients of the T aylor exp ansion of ψ at 0 ∈ R m . The algebr a of functions on N gener ate d by these c o or dinates wil l b e denote d F ( N ) , and its elements wil l b e c al le d regular functions . Explicitly , F ( N ) is generated by the functions α i j j 1 j 2 ...j r ( ψ ) = ∂ j r . . . ∂ j 1 ∂ j ψ i ( x ) | x =0 , 1 ≤ i, j, j 1 , j 2 , . . . , j r ≤ m, ψ ∈ N ; note that α i j ( ψ ) = δ i j , b ecause ψ ′ (0) = Id, while for r ≥ 1 the co eﬃ- cien ts α i j j 1 j 2 ...j r ( ψ ) are symmetric in th e lo we r indices but otherwise arbi- trary . Th us, F ( N ) can b e viewe d as the fr ee commutat ive a lgebra o v er C generated by the indeterminates { α i j j 1 j 2 ...j r ; 1 ≤ i, j, j 1 , j 2 , . . . , j r ≤ m } . The algebra F := F ( N ) inh er its from the group N a canonical Hopf algebra structure, in the standard fashion. Prop osition 2.5. With the c opr o duct ∆ : F → F ⊗ F , the antip o de S : F → F , and the c ounit ε : F → C determine d by the r e quir ements ∆( f )( ψ 1 , ψ 2 ) = f ( ψ 1 ◦ ψ 2 ) , ∀ ψ 1 , ψ 2 ∈ N , (2.22 ) S ( f )( ψ ) = f ( ψ − 1 ) , ∀ ψ ∈ N , ∀ f ∈ F , ǫ ( f ) = f ( e ) , F ( N ) is a Hopf algebr a. 41 Pr o of. The fact that these deﬁn itions give rise to a Hopf algebra is com- pletely rou tin e, once they are sh o wn to make sense. In tur n, c hec king that ∆( α i j j 1 j 2 ...j r ) ∈ F ⊗ F and S ( α i j j 1 j 2 ...j r ) ∈ F ⊗ F , (2.23) only in v olv es element ary manip ulations with th e c hain rule. F or instance, in the case of α i j k the ve riﬁcation go es as follo ws. First, for the copro du ct, ∆( α i j k )( ψ 1 , ψ 2 ) = α i j k ( ψ 1 ◦ ψ 2 ) = ∂ j ∂ k ( ψ 1 ◦ ψ 2 ) i ( x ) | x =0 = ∂ j  ( ∂ µ ψ i 1 )( ψ 2 ( x )) ∂ k ψ µ 2 ( x )  | x =0 = ( ∂ ν ∂ µ ψ i 1 )( ψ 2 ( x )) | x =0 ∂ j ψ ν 2 ( x ) | x =0 ∂ k ψ µ 2 ( x ) | x =0 + + ( ∂ µ ( ψ i 1 ( ψ 2 ( x )) | x =0 ∂ j ∂ k ψ µ 2 ( x )) | x =0 = ∂ j ∂ k ψ i 1 ( x ) | x =0 + ∂ j ∂ k ψ i 2 ( x ) | x =0 = ( α i j,k ⊗ 1 + 1 ⊗ α i j,k )( ψ 1 , ψ 2 ) , where we ha v e u sed that ψ 1 (0) = ψ 2 (0) = 0 and ψ ′ 1 (0) = ψ ′ 2 (0) = I d. T o deal with the an tip o de, one diﬀeren tiates the iden tit y ψ − 1 ( ψ ( x )) = x : δ i j = ∂ j  ( ψ − 1 ) i ( ψ ( x ))  = ∂ λ ( ψ − 1 ) i ( ψ ( x )) ∂ j ψ λ ( x ) , whic h yields under fur ther diﬀeren tiation ∂ µ ∂ λ ( ψ − 1 ) i ( ψ ( x )) ∂ k ψ µ ( x ) ∂ j ψ λ ( x ) + ∂ λ ( ψ − 1 ) i ( ψ ( x )) ∂ k ∂ j ψ λ ( x ) = 0; ev aluation at x = 0 giv es α i j k ( ψ − 1 ) + α i j k ( ψ ) = 0. T aking higher deriv ativ es one p ro v es (2.23) in a similar fashion. W e shall need an alternativ e description of the algebra F , which will b e used to recognize it as b eing iden tical to the Hopf sub algebra of H (Π) generated b y the δ i j kℓ 1 ...ℓ r ’s. Lemma 2.6. The c o eﬃci e nts of the T aylor exp ansion of ˜ ψ at e ∈ G , η i j kℓ 1 ...ℓ r ( ψ ) := γ i j kℓ 1 ...ℓ r ( ψ )( e ) , ψ ∈ N , (2.2 4) deﬁne r e gular func tions on N , w hich gener ate the algebr a F ( N ) . Pr o of. Ev aluating the expression (1.27) at e = (0 , Id ) ∈ F Π R m giv es γ i j k ℓ 1 ...ℓ r ( ψ )( e ) = ∂ ℓ r . . . ∂ ℓ 1  ( ψ ′ ( x ) − 1 ) i ν ∂ j ∂ k ψ ν ( x )  | x =0 . (2.25) 42 The deriv ativ es of ψ ′ ( x ) − 1 are sums of terms eac h of which is a pro du ct of deriv ativ es of ψ ′ ( x ) inte rsp aced w ith ψ ′ ( x ) − 1 itself. Since ψ ′ (0) = Id, the righ t h and side of (2.25) is thus seen to deﬁne a regular function on N . A more careful insp ection actuall y prov es the con v erse as w ell. First, by the v ery d eﬁnition, η i j k = α i j k . (2.26) Next, one has η i j kℓ ( ψ ) = ∂ ℓ  ( ψ ′ ( x ) − 1 ) i ν ∂ j ∂ k ψ ν ( x )  | x =0 = ∂ ℓ  ( ψ ′ ( x ) − 1 ) i ν  | x =0 ∂ j ∂ k ψ ν ( x ) | x =0 + ∂ ℓ ∂ k ∂ j ψ i ( x ) | x =0 ; on diﬀerentiat ing ( ψ ′ ( x ) − 1 ) i µ ∂ ν ψ µ ( x ) = δ i ν one sees th at ∂ ℓ  ( ψ ′ ( x ) − 1 ) i ν  | x =0 + ∂ ℓ ∂ ν ψ i ( x ) | x =0 = 0 , and therefore η i j kℓ ( ψ ) = α i j kℓ ( ψ ) − α i ℓν ( ψ ) α ν j k ( ψ ) . (2.27) By indu ction, one sho ws th at η i j kℓ 1 ...ℓ r = α i j kℓ 1 ...ℓ r + P i j kℓ 1 ...ℓ r ( α λ µν , . . . , α ρ τ σp 1 ...p r − 1 ) , (2.28) where P i j kℓ 1 ...ℓ r is a p olynomial. The triangular f orm of the identitie s (2.26)- (2.28) allo ws to reve rse the pro cess and express the α i j kℓ 1 ...ℓ r ’s in a similar fashion: α i j kℓ 1 ...ℓ r = η i j kℓ 1 ...ℓ r + Q i j kℓ 1 ...ℓ r ( η λ µν , . . . , η ρ τ σp 1 ...p r − 1 ) . (2.29) Let H (Π) ab denote the (ab elian) Hopf subalgebra of H (Π) ge nerated b y the op erators { δ i j kℓ 1 ...ℓ r ; 1 ≤ i, j, k , ℓ 1 , . . . , ℓ r ≤ m } . Prop osition 2.7. Ther e is a unique isomorphism of Hopf algebr as ι : H (Π) cop ab → F ( N ) with the pr op e rty that ι ( δ i j kℓ 1 ...ℓ r ) = η i j kℓ 1 ...ℓ r , ∀ 1 ≤ i, j, k , ℓ 1 , . . . , ℓ r ≤ m . (2.30) 43 Pr o of. In view of (2.26 )-(2.27), the generators η i j kℓ 1 ...ℓ r satisfy the analogue of the Bianc hi ident it y (1.28). Ind eed, η i j kℓ − η i j ℓk = α i j kℓ − α i ℓρ α ρ j k − α i j ℓk + α i k ρ α ρ j ℓ = η i k ρ η ρ j ℓ − η i ℓρ η ρ j k . F rom Theorem 1.9 (or rather th e pro of of Corollary 1.4) it th en f ollo ws that the assignment (2 .30 ) d o es giv e rise to a we ll-deﬁned algebra h omomorphism ι : H (Π) ab → F ( N ), which b y Lemma 2.6 is automatical ly su rjectiv e. T o pro v e th at ι : H (Π) ab → F ( N ) is injectiv e, it suﬃces to sho w that the monomials { η K ; K = in cr easingly ordered multi -index } , deﬁned in the same w a y as the δ K ’s of the P oincar ´ e-Birkhoﬀ-Witt basis of H (Π) ( cf. Prop osition 1.3), are linearly indep enden t. This can b e shown b y in duction on the heigh t. In the height 0 case the statemen t is obvious, b ecause of (2.26). Next, assum e X | J | ≤ N − 1 c J η J + X | K | = N c K η K = 0 . Using the identit ies (2.28) and (2.28), on e can replace η K b y α K + lower height . Since the α • • ... • ’s are free generators, it f ollo ws that c K = 0 f or eac h K of height N , and th us we are reduced to X | J | ≤ N − 1 c J η J = 0; the ind u ction h yp othesis no w implies c J = 0, for all J ’s. It remains to prov e that ι : H (Π) cop ab → F ( N ) is a coalgebra map, whic h amoun ts to c hec king th at ι ⊗ ι (∆ δ i j kℓ 1 ...ℓ r ) = ∆ op η i j kℓ 1 ...ℓ r . (2.31) Recall, cf. (1.51), that ∆ : H (Π) → H (Π) ⊗ H (Π) is determined by a Leibniz rule, wh ic h for δ i j kℓ 1 ...ℓ r tak es the form δ i j kℓ 1 ...ℓ r ( U ∗ φ 1 U ∗ φ 2 ) = X c iAB j δ i j A ( U ∗ φ 1 ) δ i j B ( U ∗ φ 2 ) , φ 1 , φ 2 ∈ G , whic h is equiv al ent to γ i j kℓ 1 ...ℓ r ( φ 2 ◦ φ 1 ) = X c iAB j γ i j A ( φ 1 ) γ i j B ( φ 2 ) ◦ ˜ φ 1 . (2.32) Restricting (2.32) to ψ 1 , ψ 2 ∈ N and ev aluating at e ∈ G , one obtains ∆ op η i j kℓ 1 ...ℓ r ( ψ 1 , ψ 2 ) := η i j kℓ 1 ...ℓ r ( ψ 2 ◦ ψ 1 ) = X c iAB j η i j A ( ψ 1 ) η i j B ( ψ 2 ) . 44 The righ t actio n ⊳ of G on N in duces an action of G on F ( N ), and hen ce a left action ⊲ of U ( g ) on F ( N ), deﬁn ed by ( X ⊲ f )( ψ ) = d dt | t =0 f ( ψ ⊳ exp t X ) , f ∈ F , X ∈ g . (2.33) On the other hand, there is a n atural acti on of U ( g ) on H (Π) ab , induced by the adjoint action of g on h (Π), extend ed as action b y d eriv a tions on the p olynomials in δ i j kℓ 1 ...ℓ r ’s. In order to relate these tw o actions, we need a preparatory lemma. Lemma 2.8. L et ϕ ∈ G and φ ∈ G . Then for any 1 ≤ i, j, k , ℓ 1 , . . . , ℓ r ≤ m , γ i j kℓ 1 ...ℓ r ( ϕ ◦ φ ) = γ i j kℓ 1 ...ℓ r ( φ ) , (2.34) γ i j kℓ 1 ...ℓ r ( φ ◦ ϕ ) = γ i j kℓ 1 ...ℓ r ( φ ) ◦ ˜ ϕ. (2.35 ) Pr o of. Both id en tities can b e v eriﬁed by direct compu tations, using th e explicit form ula (1.27) for γ i j kℓ 1 ...ℓ r , in conjunction with the fact that ϕ has the simple aﬃne expression ϕ ( x ) = a · x + b , a ∈ G 0 (Π), b ∈ R m . An alternativ e and more elegan t explanations r elies on the left in v ariance of the v ector ﬁelds X k , cf. Prop osition 1.1. The identit y (2.34) easily foll o ws from the co cycle prop er ty (1.16) and the fact that ϕ is aﬃn e, γ i j k ( ϕ ◦ φ ) = γ i j kℓ 1 ...ℓ r ( ϕ ) ◦ ˜ ψ + γ i j k ( φ ) = γ i j k ( φ ) , b ecause γ i j k ( ϕ ) = 0. T o c heck the second equ ation one starts with γ i j k ( φ ◦ ϕ ) = γ i j k ( φ ) ◦ ˜ ϕ + γ i j k ( ϕ ) = γ i j k ( φ ) ◦ ˜ ϕ, and notice that the inv ariance p rop erty U ϕ X U ∗ ϕ = X , for any X ∈ g , implies X  γ i j k ( φ ) ◦ ˜ ϕ  = X  γ i j k ( φ )  ◦ ˜ ϕ . W e are n o w in a p osition to formula te the precise relation b et w een th e canonical action of U ( g ) on H (Π) ab and the action ⊲ on F ( N ). Prop osition 2.9. The algebr a isomorphism ι : H (Π) ab → F ( N ) identiﬁes the U ( g ) -mo dule H (Π) ab with the U ( g ) -mo dule F ( N ) . In p ar ticular F ( N ) is U ( g ) -mo dule algebr a. 45 Pr o of. W e denote b elo w b y ϕ t the 1-parameter su bgroup exp tX of G cor- resp ond ing to X ∈ g , and emplo y the abbr eviated notation η = η i j kℓ 1 ...ℓ r , γ = γ i j kℓ 1 ...ℓ r . F rom (2.34) it follo ws that γ ( ψ ⊳ ϕ t ) = γ ( ψ ◦ ϕ t ) , whence ( X ⊲ η )( ψ ) = d dt | t =0 η ( ψ ⊳ ϕ t ) = d dt | t =0 γ ( ψ ◦ ϕ t )( e ) . No w usin g (2.35 ), one can con tinue as follo ws: d dt | t =0 γ ( ψ ◦ ϕ t )( e ) = d dt | t =0 γ ( ψ )( ˜ ϕ t ( e )) = X  γ ( ψ )  ( e ) . By iterating th is argument one obtains, for any u ∈ U ( g ), ( u ⊲ η i j kℓ 1 ...ℓ r )( ψ ) = u  γ i j kℓ 1 ...ℓ r ( ψ )  ( e ) , ψ ∈ N . (2.36) The righ t h and side of (2.36), b efore ev aluation at e ∈ G , describ es the eﬀect of the action of u ∈ U ( g ) on δ i j kℓ 1 ...ℓ r ∈ H (Π) ab . In view of the deﬁning relation (2.30) for the isomorp hism ι , this ac hieve s the p ro of. W e pro ceed to equip U ( g ) with a right F ( N )-como du le structur e. T o th is end, we assign to eac h elemen t u ∈ U ( g ) a U ( g )-v alued function on N as follo ws : ( H u )( ψ ) = ˜ u ( ψ )( e ) , where ˜ u ( ψ ) = U ψ u U ∗ ψ . (2.37 ) W e claim that H u b elongs to U ( g ) ⊗ F ( N ), and therefore the ab o v e assign- men t deﬁnes a linear map H : U ( g ) → U ( g ) ⊗ F ( N ). Indeed, let { Z I } b e the PBW basis of U ( g ) deﬁned in § 5, cf . (1.31). W e iden tify U ( g ) with the algebra of left-in v arian t diﬀerent ial op erators on G , and regard the Z I ’s as a linear b asis for these op erators. In particular, one can u niquely express U ψ Z I U ∗ ψ = X J β J I ( ψ ) Z J , ψ ∈ N , (2.38) with β J I ( ψ ) in the al gebra of functions on G generated by { γ i j K ( ψ ) } . Th e deﬁnition (2.37) then tak es th e exp licit form H Z I = X J Z J ⊗ ζ J I , where ζ J I ( ψ ) = β J I ( ψ )( e ) . (2.39 ) 46 F or example, by (1.10), (1.14) and (1.16), one has H Y i j = Y i j ⊗ 1 , (2.40) H X k = X k ⊗ 1 + Y j i ⊗ η i j k . (2.41) Th us, H : U ( g ) → U ( g ) ⊗ F ( N ) is well- deﬁn ed. Lemma 2.10. The map H : U ( g ) → U ( g ) ⊗ F ( N ) endows U ( g ) with a F ( N ) -c omo dule str uctur e. Pr o of. On th e one hand, applying (2.38) twice one obtains U ψ 1 U ψ 2 Z I U ∗ ψ 2 U ∗ ψ 1 = X J β J I ( ψ 2 ) ◦ ψ − 1 1 U ψ 1 Z J U ∗ ψ 1 = X K X J β J I ( ψ 2 ) ◦ ψ − 1 1 β K J ( ψ 1 ) ! Z K , (2.42) while on the other hand, the same left had side can b e expressed as U ψ 1 ψ 2 Z I U ∗ ψ 1 ψ 2 = X K β K I ( ψ 1 ψ 2 ) Z K ; (2.43) therefore β K I ( ψ 1 ψ 2 ) = X J β K J ( ψ 1 ) β J I ( ψ 2 ) ◦ ψ − 1 1 . (2.44) By the v ery deﬁnition (2.39) , the ident it y (2.42) giv es ( H ⊗ I d)( H Z I ) = X K Z K ⊗ X J ζ K J ⊗ ζ J I , while the d eﬁ nition (2.22) and (2.44) imply ∆ ζ K I = X J ζ K J ⊗ ζ J I . One concludes that ( H ⊗ I d)( H Z I ) = X K Z K ⊗ H ζ K I = (Id ⊗ ∆)( H Z I ) . 47 Prop osition 2.11. Equipp e d with the c o action H : U ( g ) → U ( g ) ⊗ F ( N ) , U ( g ) is a right F ( N ) - c om o dule c o algebr a. Pr o of. It is ob vious from th e deﬁnition that, for an y u ∈ U ( g ), ε ( u < 0 > ) u < 1 > = ε ( u )1 . ( 2.45) W e just hav e to c hec k that u < 0 > (1) ⊗ u < 0 > (2) ⊗ u < 1 > = u (1) < 0 > ⊗ u (2) < 0 > ⊗ u (1) < 1 > u (2) < 1 > . (2.46) In terms of the alternativ e deﬁnition (2.37), this amounts to showing that ∆  ( H u )( ψ )  = H (∆ u )( ψ ) , ∀ ψ ∈ N , (2.47) where ( H (∆ u ))( ψ ) := f ∆ u ( ψ )( e, e ) , with f ∆ u ( ψ ) = ( U ψ ⊗ U ψ ) ∆ u ( U ∗ ψ ⊗ U ∗ ψ ) . T o this end we shall use the fact that, as it follo ws f or instance fr om Prop o- sition 1.5, the decomp osition ∆ u = u (1) ⊗ u (2) is equiv alen t to the Leibniz rule u ( ab ) = u (1) ( a ) u (2) ( b ) , ∀ a, b ∈ C ∞ ( G ) . Th us, s ince ˜ u ( ψ )( ab ) = U ψ u  U ∗ ψ ( a ) U ∗ ψ ( b )  = U ψ  u (1) ( U ∗ ψ ( a )) u (2) ( U ∗ ψ ( b ))  =  U ψ u (1) U ∗ ψ  ( a ) =  U ψ u (2) U ∗ ψ  ( b ) = g u (1) ( ψ )( a ) g u (2) ( ψ )( b ); ev aluating at e ∈ G , one ob tains ( H u )( ψ )( ab ) = ( H u (1) )( ψ )( a ) ( H u (2) )( ψ )( b ) , ∀ a, b ∈ C ∞ ( G ) , whic h is tan tamoun t to (2.47). Lemma 2.12 . F or any u, v ∈ U ( g ) one has H ( uv ) = u (1) < 0 > v < 0 > ⊗ u (2) < 1 > ( u (2) ⊲ v < 1 > ) (2.48) Pr o of. Without loss of generalit y , w e ma y assum e u = Z I , v = Z J . By the deﬁnition of th e coactio n one has ˜ u ( ψ ) = U ψ u U ∗ ψ = X β K I ( ψ ) Z K , ˜ v ( ψ ) = U ψ v U ∗ ψ = β L J ( ψ ) Z L , 48 whic h yields f uv ( ψ ) = ˜ u ( ψ )( X L β L J ( ψ ) Z L ) = X L ˜ u ( ψ ) (1) ( β L J ( ψ )) ˜ u ( ψ ) (2) Z L = X L ˜ u (1) ( ψ )( β L J ( ψ )) ˜ u (2) ( ψ ) Z L ; where the last equalit y follo ws from (2.47). Denoting ˜ u (1) ( ψ ) = X M β M (1) ( ψ ) Z M , ˜ u (2) ( ψ ) = X N β N (2) ( ψ ) Z N , one can contin ue as follo ws: f uv ( ψ ) = X L,M ,N β M (1) ( ψ ) Z M ( β L J ( ψ )) β N (2) ( ψ ) Z N Z L . Ev aluat ing at e , one obtains H ( uv ) = X L,M ,N ζ M (1) ( ψ ) Z M e ( β L J ( ψ )) ζ N (2) ( ψ ) Z N e Z L e ; taking in to account that U ( g ) is co-comm utativ e, this is p recisely th e right hand side of (2.48). Lemma 2.13 . F or any u ∈ U ( g ) and any f ∈ F ( N ) one has ∆( u ⊲ f ) = u (1) < 0 > ⊲ f (1) ⊗ u (1) < 1 > ( u (2) ⊲ f (2) ) (2.49) Pr o of. By Pr op osition 2.7 we ma y assume f ∈ F ( N ) of the form f ( ψ ) = ˜ f ( ψ )( e ) , with ˜ f in the algebra generated by { γ i j kℓ 1 ...ℓ r ; 1 ≤ i, j, k , ℓ 1 , . . . , ℓ r ≤ m } . Then ∆( u ⊲ f )( ψ 1 , ψ 2 ) = ( u ⊲ f )( ψ 1 ◦ ψ 2 ) = u ( ˜ f ( ψ 1 ◦ ψ 2 ))( e ) . (2.50) No w ˜ f corresp onds to an element ˜ δ ∈ H (Π) ab , via the U ( g )-equiv arian t isomorphism ι : H (Π) cop ab → F ( N ); explicitly , ˜ δ ( g U ∗ ψ ) = ˜ f ( ψ ) g U ∗ ψ . 49 Accordingly , ˜ f ( ψ 1 ◦ ψ 2 ) U ∗ ψ 2 U ∗ ψ 1 = ˜ δ ( U ∗ ψ 2 U ∗ ψ 1 ) = ˜ δ (1) ( U ∗ ψ 2 ) ˜ δ (2) ( U ∗ ψ 1 ) = ˜ f (1) ( ψ 2 ) U ∗ ψ 2 ˜ f (2) ( ψ 1 ) U ∗ ψ 1 = ˜ f (1) ( ψ 2 )  ˜ f (2) ( ψ 1 ) ◦ ψ 2  U ∗ ψ 2 U ∗ ψ 1 , whence ˜ f ( ψ 1 ◦ ψ 2 ) = ˜ f (1) ( ψ 2 )  ˜ f (2) ( ψ 1 ) ◦ ψ 2  . Th us, we can con tin ue (2.50) as follo ws ∆( u ⊲ f )( ψ 1 , ψ 2 ) = u  ˜ f (1) ( ψ 2 ) ( ˜ f (2) ( ψ 1 ) ◦ ψ 2 )  ( e ) = u (1) ( ˜ f (1) ( ψ 2 ))( e ) u (2)  ˜ f (2) ( ψ 1 ) ◦ ψ 2 )  ( e ) = u (1) ( ˜ f (1) ( ψ 2 ))( e ) u (2)  ˜ f (2) ( ψ 1 ) ◦ ψ 2 )  ( ψ − 1 2 ( e )) = u (1) ( ˜ f (1) ( ψ 2 ))( e )  U ψ 2 u (2) U ∗ ψ 2  ( ˜ f (2) ( ψ 1 ))( e ) . Since ι sw itches the ant ip o de with its opp osite, the last line is equal to ( u (1) ⊲ f (2) )( ψ 2 ) u (2) < 1 > ( ψ 2 ) ( u (2) < 0 > ⊲ f (1) )( ψ 1 ) . Remem b ering that U ( g ) is co-comm utativ e, one ﬁnally obtains ∆( u ⊲ f )( ψ 1 , ψ 2 ) = ( u (2) ⊲ f (2) )( ψ 2 ) u (1) < 1 > ( ψ 2 ) ( u (1) < 0 > ⊲ f (1) )( ψ 1 ) . Prop osition 2.14. The H opf algebr as U := U ( g ) and F := F ( N ) form a matche d p air of H opf algebr as. Pr o of. Prop osition 2.9 together with Prop osition 2.11 sh o w that with the action and coactio n d deﬁned in (2.33) and (2.37) F is U mo dule algebra and U is a como du le coalgebra. In addition we shall s h o w that the actio n and coactio n satisfy (2.8). . . (2.12). Since U is co commutat ive and F is comm utativ e (2.12) is automatically satisﬁed. The cond itions (2.9) and (2.11) are corresp ondin gly pr o v ed in Lemma 2.13 and Lemma 2.12. Finally the conditions (2.10) and (2.8) are ob viously held. No w it is the time for th e main resu lt of this section. 50 Theorem 2.15 . The Hopf algebr as H (Π) cop and F ◮ ⊳ U ar e isomorphic. Pr o of. Prop osition 1.3 pr o vides us with δ K Z I as a b asis for th e Hopf algebra H := H (Π) cop . Let us deﬁn e I : H → F ◮ ⊳ U , b y I ( δ K Z I ) = ι ( δ K ) ◮ ⊳ Z I , wh ere ι is deﬁn ed in Prop osition 2.7, and linearly extend it on H . First let see wh y I is w ell-deﬁned. It suﬃces to sho w that the I preserve s the r elations b et w een el ements of U and F . Let X ∈ g and f ∈ F , b y usin g Prop osition 2.7, we ha v e I ( X f − f X ) = ι ( X f − f X ) ◮ ⊳ 1 = X ⊲ f ◮ ⊳ 1 = ( ι ( f ) ◮ ⊳ X ) − (1 ◮ ⊳ X )( ι ( f ) ◮ ⊳ 1) = I ( X ) I ( f ) − I ( f ) I ( X ) . No w we sho w I is in jectiv e. This can b e shown by induction on the height. In the heigh t 0 case th e statemen t is ob vious b ecause b ecause of (2.26) and the fact that α i j k ⊗ Z I is part of th e basis of F ◮ ⊳ U . Next, assume X | J | ≤ N − 1 c J,I η J ⊗ Z I + X | K | = N c K,L η K ⊗ Z L = 0 . Using the identit ies (2.28) and (2.28), on e can replace η K b y α K + lower height . Since the α • • ... • ’s are free generators, it follo ws that c K,L = 0 for eac h K of height N , and th us we are reduced to X | J | ≤ N − 1 c J,I η J ⊗ Z I = 0; the ind u ction h yp othesis no w implies c J,I = 0, for all J, I ’s. So H and F ◮ ⊳ U are isomorphic as algebras. W e n o w sho w they are isomorphic as coalgebras as well. It is enough to show I commutes with copro ducts. ∆ F ◮ ⊳ U ( I ( u )) = ∆ F ◮ ⊳ U (1 ◮ ⊳ u ) = 1 ◮ ⊳ u (1) < 0 > ⊗ u (1) < 1 > ◮ ⊳ u (2) . On the other hand let u (1) ⊗ U ϕ u (2) U ∗ ϕ = u (1) ⊗ P β I ( ϕ ) Z I . W e ha v e u ( f U ∗ ϕ g U ∗ ψ ) = u ( f g ◦ ˜ ϕ ) U ∗ ϕ U ∗ ψ = u (1) ( f ) u (2) ( g ◦ ˜ ϕ ) U ∗ ϕ U ∗ ψ = u (1) ( f ) U ∗ ϕ U ϕ u (2) U ∗ ϕ ( g ) U ∗ ψ = u (1) ( f ) U ∗ ϕ β I ( ϕ ) Z I ( g ) U ∗ ψ , 51 whic h sho ws that ∆ H cop ( u ) = u (1) ι − 1 ( u (2) < 1 > ) ⊗ u (2) < 0 > . Sin ce U ( g ) is co comm utativ e one has ( I ⊗ I )∆ H ( u ) = ∆ F ◮ ⊳ U ( I ( u )). 2.2 The non-ﬂat case W e now tak e up the case of the con tact pseudogroup Π cn , in wh ic h ca se the Kac decomp osition is giv en by Lemma 1.11. As in th e ﬂ at case, we deﬁne the c o or dinates of an elemen t ψ ∈ N cn as b eing the co eﬃcien ts of the T a ylor expansion of ψ at 0 ∈ R 2 n +1 , α i j j 1 j 2 ...j r ( ψ ) = ∂ j r . . . ∂ j 1 ∂ j ψ i ( x ) | x =0 , 0 ≤ i, j, j 1 , j 2 , . . . , j r ≤ 2 n. The algebra they generate w ill b e denoted F ( N cn ). It is the free comm uta- tiv e al gebra ge nerated b y th e indeterminates { α i j j 1 j 2 ...j r ; 0 ≤ i, j, j 1 , j 2 , . . . , j r ≤ 2 n, r ∈ R } , that are symmetric in all lo w er indices. Prop osition 2.16. F ( N cn ) is a Hopf algebr a, whose c op r o duct, antip o de and c ounit ar e unique ly determine d by the r e quir ements ∆( f )( ψ 1 , ψ 2 ) = f ( ψ 1 ◦ ψ 2 ) , ∀ ψ 1 , ψ 2 ∈ N cn , (2.51) S ( f )( ψ ) = f ( ψ − 1 ) , ∀ ψ ∈ N cn , ǫ ( f ) = f ( e ) , ∀ f ∈ F ( N cn ) . Pr o of. The pro of is almost iden tical to that of Prop osition 2.5. T here are 2 n new co ordin ates in this case, namely α i 0 , i = 1 , . . . , 2 n . for which one c hec ks that the copro duct is we ll-deﬁned as follo ws: ∆ α i 0 ( ψ 1 , ψ 2 ) = α i 0 ( ψ 1 ◦ ψ 2 ) = ∂ 0 ( ψ 1 ◦ ψ 2 ) i (0) = = ∂ 0 ψ i 1 ( ψ 2 (0)) ∂ 0 ψ 0 2 (0) + 2 n X j =1 ∂ j ψ i 1 ( ψ 2 (0)) ∂ 0 ψ j 2 (0) = = ∂ 0 ψ i 1 (0) ∂ 0 ψ 0 2 (0) + 2 n X j =1 ∂ j ψ i 1 (0) ∂ 0 ψ j 2 (0) = = ∂ 0 ψ i 1 (0) + 2 n X j =1 δ i j ∂ 0 ψ j 2 (0) = ∂ 0 ψ i 1 (0) + ∂ 0 ψ i 2 (0) = = ( α i 0 ⊗ 1 + 1 ⊗ α i 0 )( ψ 1 , ψ 2 ) . ; w e h av e b een using ab o v e the fact that, f or an y ψ ∈ N cn , ψ (0) = 0 , and ψ H ∗ (0) = Id . 52 T aking ψ 1 = ψ − 1 , ψ 2 = ψ , one obtains from the ab ov e α i 0 ( ψ − 1 ) + α i 0 ( ψ ) = 0 , h ence S α i 0 = − α i 0 . Lemma 2.17 . The c o eﬃcients of the T aylor exp a nsion of ˜ ψ at e ∈ N cn , η i j k 1 ...k r ( ψ ) := Γ i j k 1 ...k r ( ψ )( e ) , ψ ∈ N cn , (2.52) deﬁne r e gular func tions on N cn , which gener ate the algebr a F ( N cn ) . Pr o of. The formula (1.96) sho ws that η i 0 = − α i 0 , i = 1 , . . . , 2 n, while η i j = δ i j = α i j , i, j = 1 , . . . , 2 n. T o relate their higher deriv ative s, we observ e that, in view of (1.85), (1.86) X k 1 . . . X k r | (0 , Id , 1) = E k 1 . . . E k r | (0 , Id , 1) ; on the other hand, it is ob vious that the j et at 0 with resp ect to the frame { E 0 , . . . , E 2 n } is equiv ale nt to the jet at 0 with resp ect to the stand ard frame { ∂ 0 , . . . , ∂ 2 n } . This pr o v es the statemen t for the ‘new’ co ord inates. F or the other co ordinates the pro of is similar to that of Lemma 2.6. This lemma allo ws to reco v er the analog of Prop osition 2.7 b y iden tical argumen ts. Prop osition 2.18. Ther e is a u ni q ue isomorph ism of H opf algebr as ι : H (Π cn ) cop ab → F ( N cn ) with the pr o p erty that ι (∆ i j k 1 ...k r ) = η i j k 1 ...k r , 0 ≤ i, j, k 1 , k 2 , . . . , k r ≤ 2 n. Next, one has the tautologic al counterpart of Lemma 2.8. Lemma 2.19 . L et ϕ ∈ G cn and φ ∈ G cn . Then Γ i j k 1 ...k r ( ϕ ◦ φ ) = Γ i j k 1 ...k r ( φ ) , Γ i j k 1 ...k r ( φ ◦ ϕ ) = Γ i j k 1 ...k r ( φ ) ◦ ˜ ϕ. 53 Pr o of. As is the case with its sibling resu lt, this is simply a consequence of the left inv ariance the v ector ﬁ elds { X 0 , . . . , X 2 n } . In turn , the ab o ve lemma allo ws to reco v er the analog of Prop osition 2.9. Prop osition 2.20. The algebr a isomorphism ι : H (Π cn ) cop ab → F ( N cn ) iden- tiﬁes the U ( g cn ) -mo dule H (Π cn ) ab with the U ( g cn ) -mo dule F ( N cn ) . In p ar- ticular F ( N cn ) is U ( g cn ) -mo dule algebr a. F ur th ermore, U ( g cn ) can b e endo we d with a right F ( N cn )-comodu le stru c- ture H : U ( g cn ) → U ( g cn ) ⊗ F ( N cn ) in exactly the same wa y as in th e ﬂat case, cf. (2.37), Lemma 2.10, and is in fact a r igh t F ( N cn )-comodu le c oal- gebra ( c omp. Pr op . 2.11). Likewise, the analo g of Prop osition 2.14 holds true, establishing that U ( g cn ) and F ( N cn ) form a matc hed pair of Hopf alge- bras. Finally , one concludes in a similar fashion with the bicrossed pro du ct realizatio n th eorem for the conta ct case. Theorem 2.21. The Hopf algebr as H (Π cn ) cop and F ( N cn ) ◮ ⊳ U ( g cn ) ar e c ano nic al ly isomorp hic. 3 Hopf cyclic coh omology After reviewing some of the most basic n otions in Hopf cyclic cohomology , w e fo cus on the case of the Hopf algebras H (Π) constructed in the preceding section and sh ow h o w their Hopf cyclic co homology can b e reco v ered from a bico cyclic complex manufactured out of the matc hed pair. W e then illustr ate this pro cedure b y c omputin g the relativ e p erio dic Hopf c yclic cohomology of H n mo dulo gl n . F or n = 1, w e c ompletely calculate the non-p erio dized Hopf cyclic cohomolog y as well. 3.1 Quic k synopsis of Hopf cyclic cohomology Let H b e a Hopf algebra, and let C b e a left H -mo dule coalgebra, suc h that its com ultiplication and counit are H -linear, i .e. ∆( hc ) = h (1) c (1) ⊗ h (2) c (2) , ε ( hc ) = ε ( h ) ε ( c ) . W e r ecall f r om [16 ] that a right mo d ule M wh ic h is also a left como dule is called right-left stable anti-Y etter-Drinfeld mo dule (SA YD for short) o v er 54 the Hopf algebra H if it satisﬁes the f ollo wing conditions, for an y h ∈ H , and m ∈ M : m < 0 > m < − 1 > = m ( mh ) < − 1 > ⊗ ( mh ) < 0 > = S ( h (3) ) m < − 1 > h (1) ⊗ m < 0 > h (2) , where the coaction of H w as denoted by H M ( m ) = m < − 1 > ⊗ m < 0 > . Ha ving s uc h a datum ( H , C , M ), one deﬁnes ( cf. [17]) a co cyclic mo d ule { C n H ( C, M ) , ∂ i , σ j , τ } n ≥ 0 as follo ws. C n := C n H ( C, M ) = M ⊗ H C ⊗ n +1 , n ≥ 0 , with the co cyclic structure giv en by th e op erators ∂ i : C n → C n +1 , 0 ≤ i ≤ n + 1 σ j : C n → C n − 1 , 0 ≤ j ≤ n − 1 , τ : C n → C n , deﬁned explicitly as follo ws: ∂ i ( m ⊗ H ˜ c ) = m ⊗ H c 0 ⊗ . . . ⊗ ∆( c i ) ⊗ . . . ⊗ c n , ∂ n +1 ( m ⊗ H ˜ c ) = m < 0 > ⊗ H c 0 (2) ⊗ c 1 ⊗ . . . ⊗ c n ⊗ m < − 1 > c 0 (1) , σ i ( m ⊗ H ˜ c ) = m ⊗ H c 0 ⊗ . . . ⊗ ǫ ( c i +1 ) ⊗ . . . ⊗ c n , τ ( m ⊗ H ˜ c ) = m < 0 > ⊗ H c 1 ⊗ . . . ⊗ c n ⊗ m < − 1 > c 0 ; here we ha v e u s ed the abbreviation ˜ c = c 0 ⊗ . . . ⊗ c n . One c hec ks [17] that ∂ i , σ j , and τ satisfy the follo wing iden tities, whic h deﬁne the stru ctur e of a co cyclic mo dule ( cf. [5]): ∂ j ∂ i = ∂ i ∂ j − 1 , i < j, σ j σ i = σ i σ j +1 , i ≤ j (3. 1) σ j ∂ i =    ∂ i σ j − 1 i < j 1 n if i = j o r i = j + 1 ∂ i − 1 σ j i > j + 1; (3.2) τ n ∂ i = ∂ i − 1 τ n − 1 , 1 ≤ i ≤ n, τ n ∂ 0 = ∂ n (3.3) τ n σ i = σ i − 1 τ n +1 , 1 ≤ i ≤ n, τ n σ 0 = σ n τ 2 n +1 (3.4) τ n +1 n = 1 n . (3.5) 55 The motiv a ting example for the ab ov e n otion is the co cyclic complex asso ci- ated to a Hopf al gebra H end o w ed with a mo dular p air in involution , (MPI for short), ( δ , σ ), whic h w e recall from [6]. δ is an algebra map H → C , and σ ∈ H is a group -lik e elemen t, or equiv alent ly a coa lgebra map C → H . The pair ( δ , σ ) is called MPI if δ ( σ ) = 1, and ˜ S 2 δ = Adσ ; the t wisted anti p o de ˜ S δ is deﬁn ed b y ˜ S δ ( h ) = ( δ ∗ S )( h ) = δ ( h (1) ) S ( h (2) ) . One views H as a left H -mo du le coalgebra via left m ultiplication. On the other hand if one lets M = σ C δ to b e the ground ﬁ eld C endo w ed with the left H -coaction v ia σ and righ t H -action via the c haracter δ , then ( δ, σ ) is a MPI if and only if σ C δ is a SA YD. Thanks to the m ultiplication and the an tip o d e of H , one iden tiﬁes C H ( H , M ) w ith M ⊗ H ⊗ n via the m ap I : M ⊗ H H ⊗ ( n +1) → M ⊗ H ⊗ n , I ( m ⊗ H h 0 ⊗ . . . ⊗ h n ) = mh 0 (1) ⊗ S ( h (2) ) · ( h 1 ⊗ . . . ⊗ h n ) . As a result, ∂ i , σ j , and τ acquire the simpliﬁed form of the original deﬁ ni- tion [6], namely ∂ 0 ( h 1 ⊗ . . . ⊗ h n − 1 ) = 1 ⊗ h 1 ⊗ . . . ⊗ h n − 1 , ∂ j ( h 1 ⊗ . . . ⊗ h n − 1 ) = h 1 ⊗ . . . ⊗ ∆ h j ⊗ . . . ⊗ h n − 1 , 1 ≤ j ≤ n − 1 ∂ n ( h 1 ⊗ . . . ⊗ h n − 1 ) = h 1 ⊗ . . . ⊗ h n − 1 ⊗ σ, σ i ( h 1 ⊗ . . . ⊗ h n +1 ) = h 1 ⊗ . . . ⊗ ε ( h i +1 ) ⊗ . . . ⊗ h n +1 , 0 ≤ i ≤ n , τ n ( h 1 ⊗ . . . ⊗ h n ) = (∆ p − 1 e S ( h 1 )) · h 2 ⊗ . . . ⊗ h n ⊗ σ. F or completeness, w e record b elo w the bi-complex ( C C ∗ , ∗ ( C, H , M ) , b, B ) that computes the Hopf cyclic cohomology of a coalg ebra C with co eﬃcients in a SA YD mo dule M un der the symm etry of a Hopf algebra H : C C p,q ( C, H ; M ) =  C q − p H ( C, M ) , q ≥ p , 0 , q < p , where b : C n H ( C, M ) → C n +1 H ( C, M ) is giv en b y b = n +1 X i =0 ( − 1) i ∂ i ; the op er ator B : C n H ( C, M ) → C n − 1 H ( C, M ) is deﬁned by the form ula B = A ◦ B 0 , n ≥ 0 , 56 where B 0 = σ n − 1 τ (1 − ( − 1) n τ ) and A = 1 + λ + · · · + λ n , with λ = ( − 1) n − 1 τ n . The groups { H C n ( H ; δ, σ ) } n ∈ N are computed from the ﬁrst qu adrant total complex ( T C ∗ ( H ; δ, σ ) , b + B ) , T C n ( H ; δ, σ ) = n X k =0 C C k ,n − k ( H ; δ, σ ) , and the p erio d ic groups { H P i ( H ; δ, σ ) } i ∈ Z / 2 are compu ted f rom the full total complex ( T P ∗ ( H ; δ, σ ) , b + B ) , T P i ( H ; δ, σ ) = X k ∈ Z C C k ,i − k ( H ; δ, σ ) . W e n ote th at, in deﬁning th e Hopf cyclic cohomology as ab o ve , one h as the option of viewing the Hopf algebra H as a left H -mo dule coalgebra or as a righ t H -mo dule coalg ebra. It w as the ﬁrst one whic h wa s selected as the deﬁnition in [6]. T he other c hoice w ould hav e give n to the cyclic op erator the expression τ n ( h 1 ⊗ . . . ⊗ h n ) = σ S ( h n ( n ) ) ⊗ h 1 S ( h n ( n − 1) ) ⊗ . . . ⊗ h n − 1 S ( h n (1) ) δ ( S − 1 ( h n (2) )) . As it happ ens, the c hoice originally selected is not the b est suited for the situations in v olving a righ t action. T o restore the natur ality of the notation, it is then conv enien t to pass from the Hopf a lgebra H to the co -opp osite Ho pf algebra H n cop . This transition do es not aﬀect the Hopf cyclic c ohomology , b ecause for any H -mo du le coalge br a C an d an y SA YD modu le M one has a canonical equiv alence ( C ∗ H ( C, M ) , b, B ) ≃ ( C ∗ H cop ( C cop , M cop ) , b, B ); ( 3.6) M cop := M is SA YD mo du le for H cop , with the action of H cop the same as the action of H , but w ith the coaction H : M cop → H cop ⊗ M cop giv en by H ( m ) = S − 1 ( m < − 1 > ) ⊗ m < 0 > . (3.7) The equiv alence (3.6) is realized b y the map T : C n H ( C, M ) → C n H cop ( C cop , M cop ) , (3.8) T ( m ⊗ c 0 ⊗ . . . ⊗ c n ) = m < 0 > ⊗ m < − 1 > c 0 ⊗ c n ⊗ . . . ⊗ c 1 . 57 Prop osition 3.1. The map T deﬁnes an isomorphism of mixe d c ompl exes. Pr o of. The map T is well-deﬁned b ecause T ( mh ⊗ c 0 ⊗ . . . ⊗ c n ) = mh (2) ⊗ S ( h (3) ) m < − 1 > h (1) c 0 ⊗ c n ⊗ . . . ⊗ c 1 = m < 0 > ⊗ h ( n +2) S ( h ( n +3) ) m < − 1 > h (1) c 0 ⊗ h ( n +1) c n ⊗ . . . ⊗ h (2) c 1 m < 0 > ⊗ m < − 1 > h (1) c 0 ⊗ h ( n +1) c n ⊗ . . . ⊗ h (2) c 1 = T ( m ⊗ h (1) c 0 ⊗ . . . ⊗ h ( n +1) c n ) . W e denote the cyclic stru cture of C n H ( C, M ), resp. C H cop ( C cop , M cop ) by ∂ i , σ i , and τ , resp. d i , s j , and t . W e need to show that T commutes with b and B . One h as in f act a stronger comm utation prop erty , namely T ∂ i = d n +1 − i T , 0 ≤ i ≤ n + 1 . (3.9) Indeed, d n +1 T ( m ⊗ c 0 ⊗ . . . ⊗ c n ) = d n +1 ( m < 0 > ⊗ m < − 1 > c 0 ⊗ c n ⊗ . . . ⊗ c 1 ) = m < 0 > ⊗ m < − 3 > c 0 (1) ⊗ c n ⊗ . . . ⊗ c 1 ⊗ S − 1 ( m < − 1 > ) m < − 2 > c 0 (2) m < 0 > ⊗ m < − 1 > c 0 (1) ⊗ c n ⊗ . . . ⊗ c 1 ⊗ c 0 (2) T ( m ⊗ c 0 (1) ⊗ c 0 (2) ⊗ c 1 ⊗ . . . ⊗ c n ) = T ∂ 0 ( m ⊗ c 0 ⊗ . . . ⊗ c n ) T ∂ n +1 ( m ⊗ c 0 ⊗ . . . ⊗ c n ) = T ( m < 0 > ⊗ c 0 (2) ⊗ c 1 ⊗ . . . ⊗ c n ⊗ m < − 1 > c 0 (1) ) m < 0 > ⊗ m < − 1 > c 0 (2) ⊗ m < − 2 > c 0 (1) ⊗ c n ⊗ . . . ⊗ c 1 d 0 ( m < 0 > ⊗ m < − 1 > c 0 ⊗ c n ⊗ . . . ⊗ c 1 ) = d 0 T ( m ⊗ c 0 ⊗ . . . ⊗ c n ); on the other hand, for 1 ≤ i ≤ n , T ∂ i ( m ⊗ c 0 ⊗ . . . ⊗ c n ) = T ( m ⊗ c 0 ⊗ . . . ⊗ c i − 1 ⊗ c i (1) ⊗ c i (2) ⊗ c i +1 ⊗ . . . ⊗ c n ) = m < 0 > ⊗ m < − 1 > c 0 ⊗ c n ⊗ . . . ⊗ c i +1 ⊗ c i (2) ⊗ c i (1) ⊗ c i − 1 ⊗ . . . ⊗ c 1 = d n +1 − i ( m < 0 > ⊗ m < − 1 > c 0 ⊗ c n ⊗ . . . ⊗ c 1 ) = d n +1 − i T ( m ⊗ c 0 . . . c n ) . 58 Th us, T b = ( − 1) n +1 b T . Next, we c hec k that T τ = t − 1 T = t n T as follo ws: T τ ( m ⊗ c 0 ⊗ . . . ⊗ c n ) = T ( m < 0 > ⊗ c 1 ⊗ . . . ⊗ c n ⊗ m < − 1 > c 0 ) = m < 0 > ⊗ m < − 1 > c 1 ⊗ m < − 2 > c 0 ⊗ c n ⊗ . . . ⊗ c 2 = t − 1 ( m < 0 > ⊗ m < − 1 > c 0 ⊗ c n ⊗ . . . ⊗ c 1 ) = t − 1 T ( m ⊗ c 0 ⊗ . . . ⊗ c n ) . It is easy to see that T σ i = s n − 1 − i T , for 0 ≤ i ≤ n − 1. Using the ab ov e iden tities an d ts 0 = s n − 1 t 2 one obtains T B = T n − 1 X j =0 ( − 1) ( n − 1) j τ j σ n − 1 τ (1 − ( − 1) n τ ) = = n − 1 X j =0 ( − 1) ( n − ) j t − j s 0 t − 1 (1 − ( − 1) n t − 1 ) T = = n − 1 X j =0 ( − 1) ( n − 1) j t n − j t − 1 s n − 1 t 2 (1 − ( − 1) n t − 1 ) T = = ( − 1) n +1 n − 1 X j =0 ( − 1) ( n − 1) j t n − 1 − j s n − 1 t (1 − ( − 1) n t ) T = = ( − 1) n − 1 X k =0 ( − 1) ( n − 1) k t k s n − 1 t (1 − ( − 1) n t ) T = − B T , whic h completes the pr o of. W e next recall from [9] th e s etting for relativ e Hopf cyclic cohomology . Let H b e an arbitrary Hopf algebra and K ⊂ H a Hopf sub algebra. Let C = C ( H , K ) := H ⊗ K C , (3.10) where K acts on H b y right m ultiplication and on C by the counit. It is a left H -mo du le in the usual wa y , via left multiplicatio n. As su c h, it can b e identiﬁed with the qu otien t mo dule H / HK + , K + = Ker ε |K , via th e isomorphism ind uced b y h ∈ H 7− → ˙ h = h ⊗ K 1 ∈ H ⊗ K C . (3.11) 59 Moreo v er, thanks to th e right action of K on H , C = C ( H , K ) is an H -mo dule coalge bra. Indeed, its coalgebra structure is give n b y the copro duct ∆ C ( h ⊗ K 1) = ( h (1) ⊗ K 1) ⊗ ( h (2) ⊗ K 1) , (3.12) inherited from t hat on H , a nd is compatible with the action of H on C b y left multi plication: ∆ C ( g h ⊗ K 1) = ∆( g )∆ C ( h ⊗ K 1) ; similarly , there is an inherited counit ε C ( h ⊗ K 1) = ε ( h ) , ∀ c ∈ C , (3.13) that satisﬁes ε C ( g h ⊗ K 1) = ε ( g ) ε C ( h ⊗ K 1) . Th us, C is a H -mo du le coalgebra. The relativ e Hopf cyclic cohomology of H with resp ect to the K and with coeﬃcients in M , to b e denoted by H C ( H , K ; M ), is by deﬁnition the Hopf cyclic cohomolog y of C with co eﬃ- cien ts in M . Th anks to the antip o de of H one simpliﬁes the cyclic complex as follo ws ( cf. [9, § 5]): C ∗ ( H , K ; M ) = { C n ( H , K ; M ) := M ⊗ K C ⊗ n } n ≥ 0 , where K acts diagonally on C ⊗ n , ∂ 0 ( m ⊗ K c 1 ⊗ . . . ⊗ c n − 1 ) = m ⊗ K ˙ 1 ⊗ c 1 ⊗ . . . ⊗ . . . ⊗ c n − 1 , ∂ i ( m ⊗ K c 1 ⊗ . . . ⊗ c n − 1 ) = m ⊗ K c 1 ⊗ . . . ⊗ c i (1) ⊗ c i (2) ⊗ . . . ⊗ c n − 1 , ∀ 1 ≤ i ≤ n − 1 ; ∂ n ( m ⊗ K c 1 ⊗ . . . ⊗ c n − 1 ) = m (0) ⊗ K c 1 ⊗ . . . ⊗ c n − 1 ⊗ ˙ m ( − 1) ; σ i ( m ⊗ K c 1 ⊗ . . . ⊗ c n +1 ) = m ⊗ K c 1 ⊗ . . . ⊗ ε ( c i +1 ) ⊗ . . . ⊗ c n +1 , ∀ 0 ≤ i ≤ n ; τ n ( m ⊗ K ˙ h 1 ⊗ c 2 ⊗ . . . ⊗ c n ) = m (0) h 1 (1) ⊗ K S ( h 1 (2) ) · ( c 2 ⊗ . . . ⊗ c n ⊗ ˙ m ( − 1) ); the ab o ve op erators are w ell-deﬁned and endo w C ∗ ( H , K ; M ) with a cyclic structure. Since k (1) h S ( k (2) ) ⊗ K 1 = k (1) h ⊗ K ε ( k (2) ) 1 = k c ⊗ K 1 , k ∈ K , 60 the r estriction to K of the left action of H on C = H ⊗ K C can also b e regarded as ‘adjoint action’, ind uced b y conjugation. When sp ecialized to Lie algebras this d eﬁnition reco v ers the relativ e Lie algebra homology , cf. [9, Thm. 16]. I ndeed, let g b e a Lie algebra o v er the ﬁeld F , let h ⊂ g b e a reductive subalgebra in g , and let M b e a g -mo dule. W e equip M w ith the trivial g -comod ule stru cture H M ( m ) = 1 ⊗ m ∈ H ⊗ M , (3.14) and n ote the stabilit y condition is then trivially satisﬁed, while the A YD one follo ws from (3.14) and th e co comm utativit y of th e un iv ersal en v eloping algebra A ( g ). The relativ e Lie algebra h omology and cohomology of the pair h ⊂ g with co eﬃcients in M is computed from the Chev alley-Eilen b erg complexes { C ∗ ( g , h ; M ) , δ } , C n ( g , h ; M ) := M ⊗ h n ^ ( g / h ) . Here the ac tion of h on g / h is in duced by the adjoin t representat ion and the diﬀeren tials are giv en by the formulae δ ( m ⊗ h ˙ X 1 ∧ . . . ∧ ˙ X n +1 ) = n +1 X i =1 ( − 1) i +1 mX i ⊗ h ˙ X 1 ∧ . . . ∧ ˇ ˙ X i . . . ∧ ˙ X n +1 + X i ⊗ c < 0 > . W e recall that the YD condition stipulates that H C ( hc ) = h (1) c < − 1 > S ( h (3) ) ⊗ h (2) c < 0 > . (3.17 ) Using the coaction of H on C and its action on D , one constructs a co algebra structure on C ⊗ D deﬁned by the copro duct ∆( c ⊗ d ) = c (1) ⊗ c (2) < − 1 > d (1) ⊗ c (2) < 0 > ⊗ d (2) . (3.18) W e denote this coalg ebra b y C ◮ < D . The imp ort of the YD condition is rev ealed by the follo wing result. Lemma 3.2. Via the diagonal action of H , C ◮ < D b e c omes an H mo dule c o algebr a . Pr o of. One has ∆( h (1) c ◮ < h (2) d ) = h (1) c (1) ◮ < ( h (2) c (2) ) < − 1 > ( h (3) d (1) ) ⊗ ( h (2) c (2) ) < 0 > ◮ < h (4) d (2) = h (1) c (1) ◮ < h (2) c (2) < − 1 > S ( h (4) ) h (5) d (1) ⊗ h (3) c (2) < 0 > ◮ < h (6) d (2) = h (1) c (1) ◮ < h (2) c (2) < − 1 > d (1) ⊗ h (3) c (2) < 0 > ◮ < h (4) d (2) = h (1) ( c (1) ◮ < c (2) < − 1 > d (1) ) ⊗ h (2) ( c (2) < 0 > ◮ < d (2) ) = h (1) (( c ◮ < d ) (1) ) ⊗ h (2) (( c ◮ < d ) (2) ) . No w let M b e an SA YD o ve r H . W e en d o w M ⊗ C ⊗ q with the follo wing H action and coaction: ( m ⊗ ˜ c ) h = mh (1) ⊗ S ( h (2) )˜ c, H ( m ⊗ ˜ c ) = c 0 < − 1 > . . . c n < − 1 > m < − 1 > ⊗ m < 0 > ⊗ c 0 < 0 > ⊗ . . . ⊗ c n < 0 > , 62 Lemma 3.3. L et C b e an YD mo dule over H . Then via the diagonal action and c o action C ⊗ q is also an YD mo dule over H . Pr o of. W e verify that ˜ c = c 1 ⊗ . . . ⊗ c q ∈ C ⊗ n and h ∈ H satisfy (3.17). Indeed, H ( h ˜ c ) = H ( h (1) c 1 ⊗ . . . ⊗ h ( q ) c q ) = ( h (1) c 1 ) < − 1 > . . . ( h ( q ) c q ) < − 1 > ⊗ ( h (1) c 1 ) < − 1 > ⊗ . . . ⊗ ( h ( q ) c q ) < − 1 > = h (1) c 1 < − 1 > S ( h (3) ) h (4) c 2 < − 1 > S ( h (6) ) . . . h (3 q − 2) c q < − 1 > S ( h (3 q ) ) ⊗ h (2) c 1 < 0 > ⊗ h (5) c 2 < 0 > ⊗ . . . ⊗ h (3 q − 1) c q < 0 > = h (1) ˜ c < − 1 > S ( h (3) ) ⊗ h (2) ˜ c < 0 > . Prop osition 3.4. Equipp e d with the a b ove action and c o action, M ⊗ C ⊗ q is an A YD mo dule. Pr o of. Let ˜ c ∈ C ⊗ q , and h ∈ H . By Lemma 3.3, we can write H (( m ⊗ ˜ c ) h ) = H ( mh (1) ⊗ S ( h (2) )˜ c ) = ( S ( h (2) )˜ c ) < − 1 > ( mh ) < − 1 > ⊗ ( mh (1) ) < 0 > ⊗ ( S ( h (1) )˜ c ) < 0 > = S ( h (6) )˜ c < − 1 > S 2 ( h (4) ) S ( h (3) ) m < − 1 > h (1) ⊗ m < 0 > h (2) ⊗ S ( h (5) )˜ c < 0 > = S ( h (3) )˜ c < − 1 > m < − 1 > h (1) ⊗ ( m < 0 > ⊗ ˜ c < 0 > ) h (2) . W e d eﬁ ne the follo wing bigraded mo dule, inspired b y [10, 12, 1, 24], in ord er to obtain a cylindrical mo du le for cocrossed pro d u ct coa lgebras. Set X p,q := M ⊗ H D ⊗ p +1 ⊗ C ⊗ q +1 , and endow X with the op erators → ∂ i : X ( p,q ) → X ( p +1 ,q ) , 0 ≤ i ≤ p + 1 (3.19) → σ j : X ( p,q ) → X ( p − 1 ,q ) , 0 ≤ j ≤ p − 1 (3.20) → τ : X ( p,q ) → X ( p,q ) , (3.21) 63 deﬁned by → ∂ i ( m ⊗ ˜ d ⊗ ˜ c ) = m ⊗ d 0 ⊗ . . . ⊗ ∆( d i ) ⊗ . . . ⊗ d p ⊗ ˜ c, → ∂ p +1 ( m ⊗ ˜ d ⊗ ˜ c ) = m < 0 > ⊗ d 0 (2) ⊗ . . . ⊗ d p ⊗ ˜ c < − 1 > m < − 1 > d 0 (1) ⊗ ˜ c < 0 > , → σ j ( m ⊗ ˜ d ⊗ ˜ c ) = m ⊗ d 1 ⊗ . . . ⊗ ε ( d j ) ⊗ . . . ⊗ d p ⊗ ˜ c, → τ ( m ⊗ ˜ d ⊗ ˜ c ) = m < 0 > ⊗ d 1 ⊗ . . . ⊗ d p ⊗ ˜ c < − 1 > m < − 1 > d 0 ⊗ ˜ c < 0 > ; the ve rtical structure is jus t th e co cyclic structure of C ( H , K ; K ⊗ p +1 ⊗ M ), with ↑ ∂ i =: X ( p,q ) → X ( p,q +1) , 0 ≤ i ≤ q + 1 (3.2 2) ↑ σ j : X ( p,q ) → X ( p,q − 1) , 0 ≤ j ≤ q − 1 (3.23) ↑ τ : X ( p,q ) → X ( p,q ) , (3.24) deﬁned by ↑ ∂ i ( m ⊗ ˜ d ⊗ ˜ c ) = m ⊗ ˜ d ⊗ c 0 ⊗ . . . ⊗ ∆( c i ) ⊗ . . . ⊗ c q , ↑ ∂ q +1 ( m ⊗ ˜ d ⊗ ˜ c ) = = m < 0 > ⊗ S − 1 ( c 0 (1) < − 1 > ) ˜ d ⊗ c 0 (2) ⊗ c 1 ⊗ . . . ⊗ c q ⊗ m < − 1 > c 0 (1) < 0 > , ↑ σ j ( m ⊗ ˜ d ⊗ ˜ c ) = m ⊗ ˜ d ⊗ m ⊗ c 0 ⊗ . . . ⊗ ε ( c j ) ⊗ . . . ⊗ c q , ↑ τ ( m ⊗ ˜ d ⊗ ˜ c ) = m < 0 > ⊗ S − 1 ( c < − 1 > ) · ˜ d ⊗ c 1 ⊗ c 2 ⊗ . . . ⊗ c q ⊗ m < − 1 > c 0 < 0 > . Lemma 3.5. The horizontal and vertic al op er ato rs deﬁne d in (3.19) , . . . , (3.24) ar e wel l- deﬁne d and th e τ - op er ators ar e invertible. Pr o of. In view of Lemma 3.4, M ⊗ C ⊗ q is A YD mo dule and, since the q th ro w of the ab ov e bigraded complex is th e Hopf cyclic complex of K with co eﬃcien ts in M ⊗ C ⊗ q +1 , all horizon tal op erators are wel l-deﬁned [17]. By con trast, the columns are n ot Hopf cyclic m o dules of coalgebras in general, except in some sp ecial cases su c h as the case in Subs ection 3.4.1. Let us c hec k that the vertic al τ -op erator is w ell-deﬁned, whic h imp lies that 64 all the others are w ell-deﬁned. One h as ↑ τ ( mh ⊗ ˜ d ⊗ c 0 ⊗ . . . ⊗ c q ) = ( mh ) < 0 > ⊗ S − 1 ( c 0 < − 1 > ) ˜ d ⊗ c 1 ⊗ . . . ⊗ c n ⊗ ( mh ) < − 1 > c 0 < 0 > = m < 0 > h (2) ⊗ S − 1 ( c 0 < − 1 > ) ˜ d ⊗ c 1 ⊗ . . . ⊗ c n ⊗ S ( h (3) ) m < − 1 > h (1) c 0 < 0 > = m < 0 > ⊗ h (2) S − 1 ( c 0 < − 1 > ) ˜ d ⊗ h (3) c 1 ⊗ . . . ⊗ h ( n +2) c n ⊗ m < − 1 > h (1) c 0 < 0 > = m < 0 > ⊗ S − 1 (( h (2) c 0 < − 1 > S ( h (4) )) h (1) ˜ d ⊗ h (5) c 1 ⊗ . . . ⊗ h ( n +4) c n ⊗ ⊗ m < − 1 > ( h (3) c 0 ) < 0 > = m < 0 > ⊗ S − 1 (( h (2) c 0 ) < − 1 > ) h (1) ˜ d ⊗ h (3) c 1 ⊗ . . . ⊗ h ( n +2) c n ⊗ m < − 1 > ( h (2) ) < 0 > = ↑ τ ( m ⊗ h (1) ˜ d ⊗ h (2) c 0 ⊗ . . . ⊗ h ( n +2) c n ) . T o sho w that the τ -op erators are in v ertible, we write explicitl y their in v erses: ↑ τ − 1 ( m ⊗ ˜ d ⊗ ˜ c ) = m < 0 > ⊗ m < − 3 > c q < − 1 > S ( m < − 1 > ) ˜ d ⊗ m < − 2 > c q < 0 > c 0 ⊗ . . . ⊗ c q − 1 . → τ − 1 ( m ⊗ ˜ d ⊗ ˜ c ) = m < 0 > ⊗ S ( ˜ c < − 1 > m < − 1 > ) d p ⊗ d 0 ⊗ . . . ⊗ d q − 1 ⊗ ˜ c < 0 > . Using the in v ertibilit y of the antipo de of H and also the YD mo dule prop erty of C , one c hec ks that ↑ τ ◦ ↑ τ − 1 = ↑ τ − 1 ◦ ↑ τ = Id and → τ ◦ → τ − 1 = → τ − 1 ◦ → τ = Id. Prop osition 3.6. L et C b e a (c o)mo dule c o algebr a and D a mo d ule c o alge- br a over H . A ssu me that C is an Y D mo dule over H , and M is an A YD over H . Then the b i gr a de d mo dule X p,q is a cylindric al mo dule. If i n addition M ⊗ C ⊗ q is stable, then X p,q is bic o cyclic. Pr o of. Lemma 3.4 sho ws that M ⊗ C ⊗ q is A YD mo dule, and s in ce the q th ro w of the ab o v e bigraded complex is the Hopf cyclic complex of C ∗ H ( K ; M ⊗ C ⊗ q +1 ), it deﬁnes a paraco cyclic mo du le [17]. The column s are not necessarily Hopf cyclic m o dules of coalgebras though, except in some sp ecial cases. Ho wev er, one can show th at the column s are paraco cyclic mo dules. The ve riﬁcation of the fact that ↑ τ , ↑ ∂ i and ↑ σ j satisfy (3.1). . . (3.4 ) is straigh tforw ard. The only non trivial r elations are those th at inv olv e ↑ τ and ↑ ∂ p +1 , the others b eing the same as for cocyclic 65 mo dule asso ciates to coalge br as. One needs thus to prov e that ↑ τ ↑ ∂ i = ↑ ∂ i − 1 ↑ τ , 1 ≤ i ≤ q + 1 , ↑ τ ↑ ∂ 0 = ↑ ∂ q +1 , ↑ τ ↑ σ j = ↑ σ j − 1 ↑ τ , 1 ≤ j ≤ q − 1 , ↑ τ ↑ σ 0 = ↑ σ q − 1 ↑ τ 2 T o v erify these iden tities, ﬁ rst let 1 ≤ i ≤ q ; one has ↑ τ ↑ ∂ i ( m ⊗ ˜ d ⊗ c 0 ⊗ . . . ⊗ c q ) = ↑ τ ( m ⊗ ˜ d ⊗ c 0 ⊗ . . . ⊗ c i (1) ⊗ c i (2) ⊗ . . . ⊗ c q ) = m < 0 > ⊗ S − 1 ( c 0 < − 1 > ) · ˜ d ⊗ c 1 ⊗ . . . ⊗ c i (1) ⊗ c i (2) ⊗ . . . ⊗ c q ⊗ m < − 1 > c 0 < 0 > = ↑ ∂ i − 1 ( m < 0 > ⊗ S − 1 ( c 0 < − 1 > ) · ˜ d ⊗ c 1 ⊗ . . . ⊗ c q ⊗ m < − 1 > c 0 < 0 > ) = ↑ ∂ i − 1 ↑ τ ( m ⊗ ˜ d ⊗ c 0 ⊗ . . . ⊗ c q ) . Next let i = q + 1; by using the fact that C is H m o dule coalge bra, one obtains ↑ τ ↑ ∂ q +1 ( m ⊗ ˜ d ⊗ c 0 ⊗ . . . ⊗ c q ) = ↑ τ ( m < 0 > ⊗ S − 1 ( c 0 (1) < − 1 > ) ˜ d ⊗ c 0 (2) ⊗ . . . ⊗ c q ⊗ m < − 1 > c 0 (1) < 0 > ) = m < 0 > ⊗ S − 1 ( c 0 (2) < − 1 > ) S − 1 ( c 0 (1) < − 1 > ) · ˜ d ⊗ c 1 ⊗ . . . ⊗ c q ⊗ m < − 2 > c 0 (1) < 0 > ⊗ m < − 1 > c 0 (2) < 0 > = m < 0 > ⊗ S − 1 ( c 0 (1) < − 1 > c 0 (2) < − 1 > ) · ˜ d ⊗ c 1 ⊗ . . . ⊗ c q ⊗ m < − 2 > c 0 (1) < 0 > ⊗ m < − 1 > c 0 (2) < 0 > = m < 0 > ⊗ S − 1 ( c 0 < − 1 > ) · ˜ d ⊗ c 1 ⊗ . . . ⊗ c q ⊗ m < − 2 > c 0 < 0 > (1) ⊗ m < − 1 > c 0 < 0 > (2) = ↑ ∂ q ( m < 0 > ⊗ S − 1 ( c 0 < − 1 > ) · ˜ d ⊗ c 1 ⊗ . . . ⊗ c q ⊗ m < − 2 > c 0 ) = ↑ ∂ q ↑ τ ( m ⊗ ˜ d ⊗ c 0 ⊗ . . . ⊗ c q ) . Finally let i = 0; one has ↑ τ ↑ ∂ 0 ( m ⊗ ˜ d ⊗ c 0 ⊗ . . . ⊗ c q ) = ↑ τ ( m ⊗ ˜ d ⊗ c 0 (1) ⊗ c 0 (2) ⊗ c 1 ⊗ . . . ⊗ c q ) = m < 0 > ⊗ S − 1 ( c 0 (1) < − 1 > ) ˜ d ⊗ c 0 (2) ⊗ c 1 ⊗ . . . ⊗ c q ⊗ m < − 1 > c 0 (1) < 0 > = ↑ ∂ q +1 ( m ⊗ ˜ d ⊗ c 0 ⊗ . . . ⊗ c q ) . 66 The other id en tities are c hec k ed in a similar fash ion. W e next sho w that the vertic al op erators commute with h orizon tal o p erators. Using Lemma 3.5, and since in an y parco cyclic m o dule with inv ertible τ - op erator one has ∂ j = τ − j ∂ 0 τ j , 1 ≤ j ≤ n, σ i = τ − i σ n − 1 τ i , 1 ≤ i ≤ n − 1 , (3.2 5) it suﬃces to verify the id en tities ↑ τ → τ = → τ ↑ τ , ↑ τ → ∂ 0 = → ∂ 0 ↑ τ , → τ ↑ ∂ 0 = ↑ ∂ 0 → τ , ↑ τ → σ p − 1 = → σ p − 1 ↑ τ , and → τ ↑ σ q − 1 = ↑ σ q − 1 → τ . Let us chec k the ﬁr s t; one has ↑ τ → τ ( m ⊗ k 0 ⊗ . . . ⊗ k p ⊗ c 0 ⊗ . . . ⊗ c q ) = ↑ τ ( m < 0 > ⊗ k 1 ⊗ . . . ⊗ k p ⊗ ˜ c < − 1 > m < − 1 > k 0 ⊗ ˜ c < 0 > ) = m < 0 > ⊗ S − 1 ( c 0 < − 1 > ) · ( k 1 ⊗ . . . ⊗ k p ) ⊗ c 1 < − 1 > . . . c q < − 1 > m < − 2 > k 0 ⊗ c 1 < 0 > ⊗ . . . ⊗ c q < 0 > ⊗ m < − 1 > c 0 < 0 > = m < 0 > ⊗ S − 1 ( c 0 < − 2 > ) · ( k 1 ⊗ . . . ⊗ k p ) ⊗ ⊗ c 1 < − 1 > . . . c q < − 1 > m < − 4 > c 0 < − 1 > S ( m < − 2 > ) m < − 1 > S − 1 ( c 0 < − 2 > ) k 0 ⊗ ⊗ c 1 < 0 > ⊗ . . . ⊗ c q < 0 > ⊗ m < − 3 > c 0 < 0 > = m < 0 > ⊗ S − 1 ( c 0 < − 1 > ) · ( k 1 ⊗ . . . ⊗ k p ) ⊗ ⊗ c 1 < − 1 > . . . c q < − 1 > ( m < − 2 > c 0 < 0 > ) < − 1 > m < − 1 > S − 1 ( c 0 < − 2 > ) k 0 ⊗ ⊗ c 1 < 0 > ⊗ . . . ⊗ c q < 0 > ⊗ ( m < − 2 > c 0 ) < 0 > = → τ ( m < 0 > ⊗ S − 1 ( c 0 < − 1 > ) · ˜ k ⊗ c 1 ⊗ . . . ⊗ c q ⊗ m < − 1 > c 0 ) = → τ ↑ τ ( m ⊗ k 0 ⊗ . . . ⊗ k p ⊗ c 0 ⊗ . . . ⊗ c q ) Since C is a H -mo d ule coal gebra one can write ↑ τ → ∂ 0 ( m ⊗ d 0 ⊗ . . . ⊗ d p ⊗ c 0 ⊗ . . . ⊗ c q ) = ↑ τ ( m ⊗ d 0 (1) ⊗ d 0 (2) ⊗ d 1 ⊗ . . . ⊗ d p ⊗ c 0 ⊗ . . . ⊗ c q ) = ( m < 0 > ⊗ S − 1 ( c 0 < − 1 > )( d 0 (1) ⊗ d 0 (2) ⊗ d ⊗ . . . . . . ⊗ d p ) ⊗ c 1 ⊗ . . . ⊗ c q ⊗ m < − 1 > c 0 < 0 > ) = → ∂ 0 ↑ τ ( m ⊗ d 0 ⊗ . . . ⊗ d p ⊗ c 0 ⊗ . . . ⊗ c q ) . 67 T o sho w th e → τ ↑ ∂ 0 = ↑ ∂ 0 → τ one u ses only the mo d ule coalg ebra prop erty of C ; th us, → τ ↑ ∂ 0 ( m ⊗ ˜ d ⊗ ˜ c ) = → τ ( m ⊗ ˜ d ⊗ c 0 (1) ⊗ c 0 (2) ⊗ c 1 ⊗ . . . ⊗ c q ) = m < 0 > ⊗ d 1 ⊗ . . . ⊗ d p ⊗ c 0 (1) < − 1 > c 0 (2) < − 1 > c 1 < − 1 > . . . . . . c q < − 1 > m < − 1 > d 0 ⊗ c 0 (1) < 0 > ⊗ c 0 (2) < 0 > ⊗ c 1 < 0 > ⊗ . . . ⊗ c q < 0 > = m < 0 > ⊗ d 1 ⊗ . . . ⊗ d p ⊗ c 0 < − 1 > < − 1 > c 1 < − 1 > . . . . . . c q < − 1 > m < − 1 > d 0 ⊗ c 0 < 0 > (1) ⊗ c 0 < 0 > (2) ⊗ c 1 < 0 > ⊗ . . . ⊗ c q < 0 > = ↑ ∂ 0 → τ ( m ⊗ ˜ d ⊗ ˜ c ) . The r emaining relations, → τ ↑ σ q − 1 = ↑ σ q − 1 → τ and ↑ τ → σ p − 1 = → σ p − 1 ↑ τ , are ob viously tru e. Finally , by using the stabilit y of M , we v erify the cylind rical condition → τ p +1 ↑ τ q +1 = Id as follo ws: → τ p +1 ↑ τ q +1 ( m ⊗ d 0 ⊗ . . . ⊗ d p ⊗ c 0 ⊗ . . . ⊗ c q ) = → τ p +1 ( m < 0 > ⊗ S − 1 ( c 0 < − 1 > . . . c q < − 1 > ) · ( d 0 ⊗ . . . ⊗ d p ) ⊗ ⊗ m < − q − 1 > c 0 < 0 > ⊗ . . . ⊗ m < − 1 > c q < 0 > ) = m < 0 > ⊗ m < − p − q − 2 > d 0 ⊗ . . . ⊗ m < − q − 2 > d p ⊗ m < − q − 1 > c 0 ⊗ . . . ⊗ m < − 1 > c q = m ⊗ d 0 ⊗ . . . ⊗ d p ⊗ c 0 ⊗ . . . ⊗ c q . The diagonal of any cylindrical mo dule is a cocyclic mo dule [12] whose cyclic structures is give n by , ∂ i := ↑ ∂ i → ∂ i : X n,n → X n +1 ,n +1 , (3.26) σ i := ↑ σ i → ∂ i : X n,n → X n − 1 ,n − 1 , (3.27) τ := ↑ τ → τ : X n,n → X n,n . (3.28) In order to relate the Hopf cyclic cohomology of C ◮ < D to the diagonal complex of the X , we d eﬁne a map similar to the one used in [24], and use the fact th at C is a YD mo du le o ve r H to sho w that is w ell-deﬁned. Sp eciﬁcally , Ψ : C n H ( C ◮ < D ; M ) → X n,n , (3.29) Ψ( m ⊗ c 0 ◮ < d 0 ⊗ . . . ⊗ c n ◮ < d n ) = m ⊗ c 0 < − n − 1 > d 0 ⊗ . . . ⊗ c 0 < − 1 > . . . c n < − 1 > d n ⊗ c 0 < 0 > ⊗ . . . ⊗ c n < 0 > . 68 Prop osition 3.7. The map deﬁne d in (3.29) establishes a c yclic isomor- phism b etwe en the c omplex C ∗ H ( C ◮ < D ; M ) a nd the diagonal of X ∗ , ∗ . Pr o of. First w e show that the ab ov e map is w ell-deﬁned. The fact that C is YD mo d u le helps in t w o wa ys: ﬁrstly , Lemma 3.2 shows that H is acting diagonally on C ◮ < D , and secondly th e twisting ⊤ : C ⊗ D → D ⊗ C where ⊤ ( c ⊗ d ) = c < − 1 > d ⊗ c < 0 > , is H -linear; indeed, ⊤ ( h (1) c ⊗ (2) d ) = ( h (1) c ) < − 1 > ( h (2) d ) ⊗ ( h (1) ) < 0 > = h (1) c < − 1 > S ( h (3) ) h (4) d ⊗ h (2) c < 0 > = h (1) c < − 1 > d ⊗ h (2) c < 0 > . This ens u res that Ψ is w ell-deﬁned, b ecause it is obtained out of ⊤ b y iteration. In order to pr o v e that Ψ is a cyclic map, it suﬃces to c hec k that Ψ comm utes with the τ -op eratorss, th e ﬁrst co face, and the la st co degeneracy , b ecause the rest of the op erators are made of these (3.25). One v eriﬁes that Ψ comm utes with cyclic op er ators as follo ws. On the one h and, Ψ τ C ◮ ⊗ c 1 ◮ < d 1 ⊗ . . . ⊗ c n ◮ < d n ⊗ m < − 1 > c 0 ◮ < m < − 2 > d 0 ) = m < 0 > ⊗ c 1 < − n − 1 > d 1 ⊗ . . . ⊗ c 1 < − 2 > . . . c n < − 2 > d n ⊗ ⊗ c 1 < − 1 > . . . c n < − 1 > ( m < − 2 > c 0 ) < − 1 > m < − 1 > d 0 ⊗ ⊗ c 1 < 0 > ⊗ . . . ⊗ c n < 0 > ⊗ ( m < − 2 > c 0 ) < − 1 > = m < 0 > ⊗ c 1 < − n − 1 > d 1 ⊗ . . . ⊗ c 1 < − 2 > . . . c n < − 2 > d n ⊗ ⊗ c 1 < − 1 > . . . c n < − 1 > m < − 2 > c 0 < − 1 > d 0 ⊗ c 1 < 0 > ⊗ . . . ⊗ c n < 0 > ⊗ m < − 1 > c 0 < 0 > . On the other hand, → τ ↑ τ Ψ( m ⊗ c 0 ◮ < d 0 ⊗ . . . ⊗ c n ◮ < d n ) = → τ ↑ τ ( m ⊗ c 0 < − n − 1 > d 0 ⊗ . . . ⊗ c 0 < − 1 > . . . c n < − 1 > d n ⊗ c 0 < 0 > ⊗ . . . ⊗ c n < 0 > ) = → τ ( m < 0 > ⊗ c 0 < − n − 1 > c 1 < − n − 1 > d 1 ⊗ . . . ⊗ c 0 < − 2 > . . . c n < − 2 > d n ⊗ c 0 < − 1 > . . . c n < − 1 > m < − 1 > c 0 < − n − 2 > d 0 ⊗ c 0 < 0 > ⊗ . . . ⊗ c n < 0 > ) = m < 0 > ⊗ S − 1 ( c 0 < − 1 > ) · ( c 0 < − n − 1 > c 1 < − n − 1 > d 1 ⊗ . . . ⊗ c 0 < − 2 > . . . c n < − 2 > d n ⊗ c 0 < − 1 > . . . c n < − 1 > m < − 1 > c 0 < − n − 2 > d 0 ) ⊗ c 1 < 0 > ⊗ . . . ⊗ c n < 0 > ⊗ m < − 1 > c 0 < 0 > = m < 0 > ⊗ c 1 < − n − 1 > d 1 ⊗ . . . ⊗ c 1 < − 2 > . . . c n < − 2 > d n ⊗ ⊗ c 1 < − 1 > . . . c n < − 1 > m < − 2 > c 0 < − 1 > d 0 ⊗ c 1 < 0 > ⊗ . . . ⊗ c n < 0 > ⊗ m < − 1 > c 0 < 0 > . 69 The next to chec k is the equalit y ∂ 0 Ψ = Ψ ∂ 0 C ◮ d 0 ⊗ . . . ⊗ c 0 < − 1 > . . . c n < − 1 > d n ⊗ c 0 < 0 > ⊗ . . . ⊗ c n < 0 > ) = m ⊗ c 0 < − n − 2 > d 0 (1) ⊗ c 0 < − n − 1 > d 0 (2) ⊗ . . . ⊗ c 0 < − 1 > . . . c n < − 1 > d n ⊗ c 0 < 0 > (1) ⊗ c 0 < 0 > (2) ⊗ c 1 < 0 > ⊗ . . . ⊗ c n < 0 > = m ⊗ c 0 (1) < − n − 2 > c 0 (2) < − n − 2 > d 0 (1) ⊗ c 0 (1) < − n − 1 > c 0 (2) < − n − 1 > d 0 (2) ⊗ . . . ⊗ c 0 (1) < − 1 > c 0 (2) < − 1 > c 1 < − 1 > . . . c n < − 1 > d n ⊗ c 0 (1) < 0 > ⊗ c 0 (2) < 0 > ⊗ c 1 < 0 > ⊗ . . . . . . ⊗ c n < 0 > . On the other hand, Ψ ∂ 0 C ◮ d 0 (1) ⊗ c 0 (2) < 0 > ◮ < d 0 (2) ⊗ c 1 ◮ < d 1 ⊗ . . . . . . ⊗ c n ◮ < d n ) = m ⊗ c 0 (1) < − n − 2 > c 0 (2) < − 1 > d 0 (1) ⊗ c 0 (1) < − n − 1 > c 0 (2) < 0 > < − n − 1 > d 0 (2) ⊗ c 0 (1) < − n> c 0 (2) < 0 > < − n> c 1 < − n> d 1 ⊗ . . . ⊗ c 0 (1) < − 1 > c 0 (2) < 0 > < − 1 > . . . c n < − 1 > d n ⊗ c 0 (1) < 0 > ⊗ c 0 (2) < 0 > < 0 > ⊗ c 1 < 0 > ⊗ . . . ⊗ c n < 0 > = m ⊗ c 0 (1) < − n − 2 > c 0 (2) < − n − 2 > d 0 (1) ⊗ c 0 (1) < − n − 1 > c 0 (2) < − n − 1 > d 0 (2) ⊗ . . . ⊗ c 0 (1) < − 1 > c 0 (2) < − 1 > c 1 < − 1 > . . . c n < − 1 > d n ⊗ c 0 (1) < 0 > ⊗ c 0 (2) < 0 > ⊗ c 1 < 0 > ⊗ . . . . . . ⊗ c n < 0 > . The equalit y σ n − 1 Ψ = Ψ σ n − 1 C ◮ ⊗ S − 1 ( c < − 1 > ) d . It is easy to see that ⊤ and ⊥ are in v erse to one another. Ex p licitly the inv erse of Ψ is deﬁned by Ψ − 1 ( m ⊗ d 0 ⊗ . . . ⊗ d n ⊗ c 0 ⊗ . . . ⊗ c n ) = = m ⊗ c 0 < 0 > ◮ < S − 1 ( c 0 < − 1 > ) d 0 ⊗ . . . . . . ⊗ c n < 0 > ◮ < S − 1 ( c 0 < − n − 1 > c 1 < − n> . . . c n < − 1 > ) d n Applying n o w the cyclic version of the Eilenberg-Zilb er theorem [12] one obtains the sough t-for quasi-isomorphism of mixed complexes. 70 Prop osition 3.8. The mixe d c omplexes ( C ∗ H ( D ◮ < C, M ) , b, B ) and (T ot ( X ) , b T , B T )) ar e quasi-isomorp hic. The r est o f this sec tion is devote d to sho wing th at the Hopf algebras that mak e the ob ject of this pap er satisfy the conditions of the ab ov e prop osition. T o put this in the p rop er sett ing, w e let H b e a Hopf alge bra, and we consider a pair of H -mo dule coalge bras C , D such that C is H -mo dule coalge br a and via its action and coaction it is an YD mo dule o v er H . Let L ⊂ K b e Hopf subalgebras of H . O ne deﬁnes the coalgebra C := H ⊗ K C , where K acts on H by multiplicati on and on C via counit (cf [9, § 5]) . In the same fashion one deﬁn es the coalgebra K L := K ⊗ L C . If h ∈ H and c = ˙ h b e its class in C , then ∆( c ) = c (1) ⊗ c (2) := ˙ h (1) ⊗ ˙ h (2) , ε ( c ) := ε ( h ) This coalgebra h as a natural coac tion from H . H ( c ) = c < − 1 > ⊗ c < 0 > := h (1) S ( h (3) ) ⊗ ˙ h (2) . Lemma 3.9. The ab ove action and c o action ar e wel l-deﬁne d and make C a (c o) mo dule c o algebr a , r esp e ctively. In add ition b oth make C an YD mo dule over H . Pr o of. First let us c hec k that the acti on and coact ion are we ll deﬁned. With h, g ∈ H , and k ∈ K , one has h ( g k ⊗ 1) = hg k ⊗ K 1 = hg ⊗ K ǫ ( k ) = h ( g ⊗ ǫ ( k )) , whic h veriﬁes the claim for the action. F or the coaction, we write H ( hk ⊗ 1) = h (1) k (1) S ( k (3) ) S ( h (3) ) ⊗ ( h (2) k (2) ⊗ K 1) = h (1) k (1) S ( k (3) ) S ( h (3) ) ⊗ ( h (2) ⊗ K ǫ ( k (2) )) = = h (1) k (1) S ( k (2) ) S ( h (3) ) ⊗ ( h (2) ⊗ K 1) = h (1) S ( h (3) )) ⊗ ( h (2) ⊗ K ǫ ( k )) = H ( h ⊗ ǫ ( k )) . It is ob vious that these are in d eed actio n, resp . coactio n, and th us deﬁne a mo du le coal gebra structur e, resp. a mo du le coa lgebra structure o v er C . One c hec ks that the action and coaction s atisfy (3.17) as follo ws: H ( h ( ˙ g ) = H ( ˙ z}|{ hg ) = h (1) g (1) S ( g (3) ) S ( h (3) ) ⊗ ˙ z }| { h (2) g (2) = h (1) ( g (1) S ( g (3) )) S ( h (3) ) ⊗ h (2) ˙ g (2) . 71 Let u s assum e th at H acts on K , as well and via th is action K is a m o dule coalge bra. S imilarly to C in relation to H , K L inherits in a natural w a y an action from H . This mak es K L a m o dule coalge bra o v er H . Precisely , if h ∈ H and k ∈ K , den oting k ⊗ L 1 b y ˙ k , one d eﬁ nes h · ˙ k = hk ⊗ L 1 = ˙ z}|{ hk . No w C and K L together w ith their action and coaction satisfy all condi- tions of Prop osition 3.6. As a result one can form the cylind rical mo du le X ( H , C , K L ; M ). Due to the sp ecial prop erties d iscussed ab o v e, w e ma y exp ect some simpliﬁcation. Ind eed, let Y p,q = M ⊗ K K ⊗ p +1 L ⊗ C ⊗ q W e deﬁne the follo wing map from X to Y : Φ 1 : X p,q → Y p,q , (3.30) Φ 1 ( m ⊗ H ˜ ˙ k ⊗ ˙ h 0 ⊗ . . . ⊗ ˙ h n ) = mh 0 (2) ⊗ K S − 1 ( h 0 (1) ) · ˜ ˙ k ⊗ S ( h 0 (3) ) · ( ˙ h 1 ⊗ . . . ⊗ ˙ h q ) . Lemma 3.10 . The map Φ 1 deﬁne d in (3.30) i s a wel l-deﬁne d isomorphisms of ve ctor sp ac es. Pr o of. T o chec k that it is well- deﬁn ed, we clarify the ambiguities in the deﬁnition of Φ 1 as follo ws. L et k ∈ K , ˜ ˙ k ∈ K ⊗ ( p +1) L , and g , h 0 . . . h q ∈ H ; then Φ 1 ( m ⊗ ˜ ˙ k ⊗ ( h 0 k ⊗ K 1) ⊗ ˙ h 1 ⊗ . . . ⊗ ˙ h q )) = mh 0 (2) k (2) ⊗ K S − 1 ( h 1 (1) k (1) ) · ˜ ˙ k ⊗ S ( h 0 (3) k (3) ) · ( ˙ h 1 ⊗ . . . ⊗ ˙ h q ) = mh 0 (2) ⊗ K k (2) S − 1 ( k (1) ) S − 1 ( h 1 (1) ) · ˜ ˙ k ⊗ k (3) S ( k (4) ) S ( h 0 (3) ) · ( ˙ h 1 ⊗ . . . ⊗ ˙ h q ) = Φ 1 ( m ⊗ ˜ ˙ k ⊗ ( h 0 ⊗ K ǫ ( k )) ⊗ ˙ h 1 ⊗ . . . ⊗ ˙ h n )) . 72 Also, Φ 1 ( m ⊗ g (1) · ˜ ˙ k ⊗ g (2) · ( ˙ h 0 ⊗ ˙ h 1 ⊗ . . . ⊗ ˙ h q )) = m ⊗ K g (1) · ˜ ˙ k ⊗ g (2) · ( ˙ h 0 ⊗ ˙ h 1 ⊗ . . . ⊗ ˙ h q ) = mg (3) h 0 (2) ⊗ K S − 1 ( h 0 (1) ) S − 1 ( g (2) ) g (1) · ˜ ˙ k ⊗ ⊗ S ( h 0 (3) ) S ( g (4) ) g (5) · ( ˙ h 1 ⊗ . . . ⊗ ˙ h q ) = mg h 0 (2) ⊗ K S − 1 ( h 0 (1) ) · ˜ ˜ k ⊗ S ( h 0 (3) ) · ( ˙ h 1 ⊗ . . . ⊗ ˙ h q ) = Φ 1 ( mg ⊗ ˜ ˙ k ⊗ ˙ h 0 ⊗ ˙ h 1 ⊗ . . . ⊗ ˙ h q ) . One easily c hec ks th at the follo wing map deﬁnes an inv erse for Φ 1 : Φ − 1 1 : Y p,q → X p,q , Φ − 1 1 ( m ⊗ K ˜ ˙ k ⊗ ˜ ˙ h ) = m ⊗ H ˜ ˙ k ⊗ ˙ 1 ⊗ ˜ ˙ h. W e n ext push forward th e cylind r ical structure of X to get th e follo wing cylindrical structur e on Y : → ∂ i ( m ⊗ ˜ ˙ k ⊗ ˜ ˙ h ) = m ⊗ ˙ k 0 ⊗ . . . ⊗ ∆( ˙ k i ) ⊗ . . . ⊗ ˙ k p ⊗ ˜ ˙ h, → ∂ p +1 ( m ⊗ ˜ ˙ k ⊗ ˜ ˙ h ) = m < 0 > ⊗ ˙ k 0 (2) ⊗ . . . ⊗ ˙ k p ⊗ ˜ ˙ h < − 1 > m < − 1 > ˙ k 0 (1) ⊗ ˜ ˙ h < 0 > , → σ j ( m ⊗ ˜ ˙ k ⊗ ˜ ˙ h ) = m ⊗ ˙ k 1 ⊗ . . . ⊗ ε ( ˙ k j ) ⊗ . . . ⊗ ˙ k p ⊗ ˜ ˙ h, → τ ( m ⊗ ˜ ˙ k ⊗ ˜ ˙ h ) = m < 0 > ⊗ ˙ k 1 ⊗ . . . ⊗ ˙ k p ⊗ ˜ ˙ h < − 1 > m < − 1 > ˙ k 0 ⊗ ˜ ˙ h < 0 > ↑ ∂ 0 ( m ⊗ ˜ ˙ k ⊗ ˜ ˙ h ) = m ⊗ ˜ ˙ k ⊗ ˙ 1 ⊗ ˙ h 1 ⊗ . . . ⊗ ˙ h q , ↑ ∂ i ( m ⊗ ˜ ˙ k ⊗ ˜ ˙ h ) = m ⊗ ˜ ˙ k ⊗ ˙ h 0 ⊗ . . . ⊗ ∆( ˙ h i ) ⊗ . . . ⊗ ˙ h q , ↑ ∂ q +1 ( m ⊗ ˜ ˙ k ⊗ ˜ ˙ h ) = m < 0 > ⊗ ˜ ˙ k ⊗ ˙ h 1 ⊗ . . . ⊗ ˙ h q ⊗ m < − 1 > , ↑ σ j ( m ⊗ ˜ ˙ k ⊗ ˜ ˙ h ) = m ⊗ ˜ ˙ k ⊗ m ⊗ ˙ h 1 ⊗ . . . ⊗ ε ( ˙ h j +1 ) ⊗ . . . ⊗ ˙ h q , ↑ τ ( m ⊗ ˜ ˙ k ⊗ ˜ ˙ h ) = m < 0 > h 1 (2) ⊗ S − 1 ( h 1 (1) ) · ˜ ˙ k ⊗ S ( h 1 (3) ) · ( ˙ h 2 ⊗ . . . ⊗ ˙ h q ⊗ m < − 1 > ) . T o obtain a f u rther simpliﬁcation, w e assume that the action of K ⊂ H on K coincides with the multiplicati on by K . W e deﬁne a map f rom Y p,q to Z p,q := Z p,q ( H , K , L ; M ) := M ⊗ L K ⊗ p L ⊗ C ⊗ q , 73 as follo ws: Φ 2 : Y p,q → Z p,q , (3.31) Φ 2 ( m ⊗ K ˙ k 0 ⊗ . . . ⊗ ˙ k p ⊗ ˜ ˙ h ) = mk 0 (1) ⊗ L S ( k 0 (2) ) · ( ˙ k 1 ⊗ . . . ⊗ ˙ k p ⊗ ˜ ˙ h ) . Lemma 3.11. The map Φ 2 deﬁne d in (3.31) is a wel l-deﬁne d isomorph ism of ve ctor sp ac es. Pr o of. One has Φ 2 ( m ⊗ k · ( ˙ k 0 ⊗ . . . ⊗ ˙ k p ⊗ ˜ ˙ h )) = Φ 2 ( m ⊗ k (1) · ˙ k 0 ⊗ k (2) · ( ˙ k 1 ⊗ . . . ⊗ ˙ k p ⊗ ˜ ˙ h )) = mk (1) ˙ k 0 (1) ⊗ S ( k (2) ˙ k 0 (2) ) · ( k (3) · ( ˙ k 2 ⊗ . . . ⊗ ˙ k p ⊗ ˜ ˙ h )) = mk (1) ǫ ( k (2) ˙ k 0 (1) ) ⊗ S ( ˙ k 0 (2) ) · ( ˙ k 2 ⊗ . . . ⊗ ˙ k p ⊗ ˜ ˙ h ) = Φ 2 ( mk ⊗ ˙ k 0 ⊗ . . . ⊗ ˙ k p ⊗ ˜ ˙ h ) . The inv erse of Φ 2 is giv en b y Φ − 1 2 : Z p,q → Y p,q , Φ − 1 2 ( m ⊗ ˙ k 1 ⊗ . . . ⊗ ˙ k p ⊗ ˙ h 1 ⊗ . . . ⊗ ˙ h q ) = m ⊗ K ˙ 1 ⊗ ˙ k 1 ⊗ . . . ⊗ ˙ k p ⊗ ˙ h 1 ⊗ . . . ⊗ ˙ h q . W e now p ush forwa rd the cylindrical structur e of of Y on Z to get the follo wing op erators on Z ∗ , ∗ : → ∂ 0 ( m ⊗ ˜ ˙ k ⊗ ˜ ˙ h ) = m ⊗ ˙ 1 ⊗ ˙ k 1 ⊗ . . . ⊗ ˙ k p ⊗ ˜ ˙ h, → ∂ i ( m ⊗ ˜ ˙ k ⊗ ˜ ˙ h ) = m ⊗ ˙ k 1 ⊗ . . . ⊗ ∆( ˙ k i ) ⊗ . . . ⊗ ˙ k p ⊗ ˜ ˙ h, → ∂ p +1 ( m ⊗ ˜ ˙ k ⊗ ˜ ˙ h ) = m < 0 > ⊗ ˙ k 1 ⊗ . . . ⊗ ˙ k p ⊗ ˜ ˙ h < − 1 > ˙ z }| { m < − 1 > ⊗ ˜ ˙ h < 0 > , → σ j ( m ⊗ ˜ ˙ k ⊗ ˜ ˙ h ) = m ⊗ ˙ k 1 ⊗ . . . ⊗ ε ( ˙ k j +1 ) ⊗ . . . ⊗ ˙ k p ⊗ ˜ ˙ h, → τ ( m ⊗ ˜ ˙ k ⊗ ˜ ˙ h ) = m < 0 > k 1 (1) ⊗ S ( k 1 (2) ) · ( ˙ k 2 ⊗ . . . ⊗ ˙ k p ⊗ ˜ ˙ h < − 1 > ˙ z }| { m < − 1 > ⊗ ˜ ˙ h < 0 > ) . 74 ↑ ∂ 0 ( m ⊗ ˜ ˙ k ⊗ ˜ ˙ h ) = m ⊗ ˜ ˙ k ⊗ ˙ 1 ⊗ ˙ h 1 ⊗ . . . ⊗ ˙ h q , ↑ ∂ i ( m ⊗ ˜ ˙ k ⊗ ˜ ˙ h ) = m ⊗ ˜ ˙ k ⊗ ˙ h 0 ⊗ . . . ⊗ ∆( ˙ h i ) ⊗ . . . ⊗ ˙ h q , ↑ ∂ q +1 ( m ⊗ ˜ ˙ k ⊗ ˜ ˙ h ) = m < 0 > ⊗ ˜ ˙ k ⊗ ˙ h 1 ⊗ . . . ⊗ ˙ h q ⊗ ˙ z }| { m < − 1 > , ↑ σ j ( m ⊗ ˜ ˙ k ⊗ ˜ ˙ h ) = m ⊗ ˜ ˙ k ⊗ ˙ h 1 ⊗ . . . ⊗ ε ( ˙ h j +1 ) ⊗ . . . ⊗ ˙ h q , ↑ τ ( m ⊗ ˜ ˙ k ⊗ ˜ ˙ h ) = m < 0 > h 1 (4) S − 1 ( h 1 (3) · 1 K ) ⊗ S ( S − 1 ( h 1 (2) ) · 1 K ) ·  S − 1 ( h 1 (1) ) · ˜ ˙ k ⊗ S ( h 1 (5) ) · ( ˙ h 2 ⊗ . . . ⊗ ˙ h q ⊗ ˙ z }| { m < − 1 > )  . Lemma 3.12 . The ab o ve op er at ors deﬁne d on Z ar e wel l-deﬁne d and yield a cylindric al mo dule. Pr o of. The second part of the lemma holds b y the v ery deﬁnition, in view of the f act that X is cylind rical m o dule. W e chec k the ﬁ rst cla im for ↑ τ and → τ , for th e other op erators b eing obviously tru e. Since M is A YD and C is Y D , w e ha ve → τ ( m ⊗ l · ( ˜ ˙ k ⊗ ˜ ˙ h )) = m < 0 > l (1) k 1 (1) ⊗ L S ( l (2) k 1 (2) ) · ( l (3) ˙ k 2 ⊗ . . . ⊗ l ( p +1) ˙ k p ⊗ z }| { ( l ( p +2) ˜ ˙ h ) < − 1 > m < − 1 > ⊗ ( l ( p +2) ˜ ˙ h ) < 0 > ) = m < 0 > l (1) k 1 (1) ⊗ L S ( l (2) k 1 (2) ) · ( l (3) ˙ k 2 ⊗ . . . ⊗ l ( p +1) ˙ k p ⊗ z }| { l ( p +2) ˜ ˙ h < − 1 > S ( l ( p +4) ) m < − 1 > ⊗ l ( p +3) ˜ ˙ h < 0 > ) = m < 0 > l (1) k 1 (1) ⊗ L S ( k 1 (2) ) · ( ˙ k 2 ⊗ . . . ⊗ ˙ k p ⊗ z }| { ˜ ˙ h < − 1 > S ( l (2) ) m < − 1 > ⊗ ˜ ˙ h < 0 > ) = m < 0 > l (2) k 1 (1) ⊗ L S ( k 1 (2) ) · ( ˙ k 2 ⊗ . . . ⊗ ˙ k p ⊗ z }| { ˜ ˙ h < − 1 > S ( l (3) ) m < − 1 > l (1) ⊗ ˜ ˙ h < 0 > ) = → τ ( ml ⊗ ˜ ˙ k ⊗ ˜ ˙ h ) . 75 F or the ve rtical cyclic op erator one only u ses the A YD prop ert y of M ; th us, ↑ τ ( m ⊗ l · ( ˜ ˙ k ⊗ ˜ ˙ h )) = m < 0 > l (5) h 1 (4) S − 1 ( l (4) h 1 (3) · 1 K ) ⊗ L S ( S − 1 ( l (3) h 1 (2) ) · 1 K ) ·  S − 1 ( l (2) h 1 (1) ) · l (1) ˜ ˙ k ⊗ S ( l (6) h 1 (5) ) · ( l (7) ˙ h 2 ⊗ . . . ⊗ l (5+ q ) ˙ h q ⊗ ˙ m < − 1 > )  = m < 0 > l (1) h 1 (4) S − 1 ( h 1 (3) · 1 K ) ⊗ L S ( S − 1 ( h 1 (2) ) · 1 K ) ·  S − 1 ( h 1 (1) ) · ˜ ˙ k ⊗ S ( h 1 (5) ) · ( ˙ h 2 ⊗ . . . . . . ⊗ ˙ h q ⊗ S ( l (2) ) ˙ z }| { m < − 1 > )  = m < 0 > l (2) h 1 (4) S − 1 ( h 1 (3) · 1 K ) ⊗ L S ( S − 1 ( h 1 (2) ) · 1 K ) ·  S − 1 ( h 1 (1) ) · ˜ ˙ k ⊗ ⊗ S ( h 1 (5) ) · ( ˙ h 2 ⊗ . . . ⊗ ˙ h q ⊗ ˙ z }| { S ( l (3) ) m < − 1 > l (1) ) ! = ↑ τ ( ml ⊗ ˜ ˙ k ⊗ ˜ ˙ h ) . Prop osition 3.13. The fol lowing map deﬁnes an isom orphism of c o cyclic mo dules, Θ := Φ 2 ◦ Φ 1 ◦ Ψ : C ∗ H ( C ◮ ⊳ K L ; M ) → Z ∗ , ∗ L , (3.32) wher e Ψ , Φ 1 , and Φ 2 ar e d eﬁne d in (3.29) , (3.30) and (3.31 ) r esp e ctively. Pr o of. The maps Φ 2 , Φ 1 , and Ψ are isomorphisms. Deﬁne now the map Φ : H ⊗ L 1 → C ◮ < K L , by the form ula Φ( h ⊗ L 1) = ˙ z}|{ h (1) ◮ < ˙ z }| { h (2) · 1 K = ( h (1) ⊗ K 1) ◮ < ( h (2) · 1 K ⊗ L 1) . (3.33) Prop osition 3.14. The map Φ is a map of H -mo dule c o algebr as. Pr o of. Using that if h ∈ H and l ∈ L then ( hl ) · 1 K = ( h · 1 K ) l , one c hec ks that Φ is w ell-deﬁned, as f ollo ws: Φ( hl ⊗ 1) = ( h (1) l (1) ⊗ K 1) ◮ < ( h (2) l (2) · 1 K ⊗ L 1) = ( h (1) ⊗ K ǫ ( l (1) )) ◮ < (( h (2) · 1 K ) l (2) ⊗ L 1) = ( h (1) ⊗ K ǫ ( l (1) )) ◮ < (( h (2) · 1 K ) ⊗ L ǫ ( l (2) )) = Φ( h ⊗ ǫ ( l )) . 76 Next, us in g the coalge br a s tructure of C ◮ < K L , one can w rite ∆(Φ( h )) = ∆( ˙ z}|{ h (1) ◮ < ˙ z }| { h (2) · 1 K ) = ˙ z}|{ h (1) ◮ ⊳ ˙ z }| { h (2) S ( h (4) ) · ( h (5) · 1 K ) ⊗ ˙ z}|{ h (3) ◮ < ˙ z }| { h (6) · 1 K = ˙ z}|{ h (1) ◮ ⊳ ˙ z }| { ( h (2) S ( h (4) ) h (5) ) · 1 K ⊗ ˙ z}|{ h (3) ◮ < ˙ z }| { h (6) · 1 K = ˙ z}|{ h (1) ◮ ⊳ ˙ z }| { h (2) · 1 K ⊗ ˙ z}|{ h (3) ◮ < ˙ z }| { h (4) · 1 K = Φ( h (1) ) ⊗ Φ( h (2) ) . Finally , the H -linearit y is trivial b ecause H ac ts on C ◮ < K L diagonally . Summing up , we conclude with th e follo wing r esult, whic h applies to a many cases of in terest, in particular to those w hic h mak e the main ob ject of this pap er. Theorem 3.15. A ssuming that Φ i s an isomorphism, the Hopf c yclic c om- plex of the Hopf algebr a H r elative to the Hopf sub algebr a L with c o eﬃcients in SA YD M is quasi-isomorphic with the total c omplex of the mixe d c ompl ex of Z ( H , K , L ; M ) . As a matter of fact, the cylindrical mo du les Z and Y are often bico cyclic mo dules, for example if the SA YD M h as th e prop erty that m < − 1 > ⊗ m < 0 > ∈ K ⊗ M , ∀ m ∈ M ; in this case M will b e called K -S A YD. In this pap er we compu te Hopf cyclic cohomology with co eﬃcien ts in C δ , wh ic h is ob viously K -SA YD for an y Hopf subalgebra K ⊂ H . Prop osition 3.16. If M is K - SA YD mo dule then M ⊗ C ⊗ q is SA YD, and henc e Z and Y ar e bic o cyclic mo dules. Pr o of. Only the stabilit y cond ition remains to b e pr o v ed. W e c hec k it only for q = 1, bu t the same pr o of w orks for all q ≥ 1. By the v ery deﬁnition of the coaction, ∆( m ⊗ h ) = h (1) S ( h (3) ) m < − 1 > ⊗ m < 0 > ⊗ h (2) W e need to verify the iden tit y ( m ⊗ h ) < 0 > ( m ⊗ h ) < − 1 > = m ⊗ h. 77 The left han d side can b e expressed as follo ws: ( m ⊗ h ) < 0 > ( m ⊗ h ) < − 1 > = m < 0 > h (1) S ( h (5) ) m < − 2 > ⊗ S ( m < − 1 > ) S 2 ( h (4) ) S ( h (2) ) h (3) = m < 0 > h (1) S ( h (3) ) m < − 2 > ⊗ S ( m < − 1 > ) S 2 ( h (2) ) . W e now r ecall that the c yclic op erator τ 1 : M ⊗ H → M ⊗ H is giv en by τ 1 ( m ⊗ h ) = m < 0 > h (1) ⊗ S ( h (2) ) m < − 1 > . S ince τ 2 1 ( m ⊗ h ) = τ 1 ( m < − 1 > h (1) ⊗ S ( h (1) ) m < − 1 > ) = m < 0 > h (2) S ( h (5) ) m < − 3 > ⊗ S ( m < − 2 > ) S 2 ( h (4) ) S ( h (3) ) m < − 1 > h (1) , the desired equalit y simply f ollo ws f rom the facts that τ 2 1 = Id and M is K -SA YD. 3.3 Bico cyclic c omplex for primitiv e Hopf algebras W e n ow pro ceed to s h o w that the Hopf algebras H (Π) asso ciated to a primi- tiv e pseudogroups do satisfy all the requ ir emen ts o f the preceding subsection, and so Theorem 3.15 applies to allo w the computation of their Hopf cyclic cohomology by means of a b ico cyclic complex. First w e note that, in view of Prop osition 3.1, we can rep lace H (Π) by H (Π) cop . By Theorems 2.15 and 2.21, th e latter can b e identiﬁed to the bicrossed pro du ct F ◮ ⊳ U , where U := U ( g ), F := F ( N ), and we shall d o so from n o w on without further wa rnin g. Next, we need b oth F and U to b e H -mo du le coalge bras. Recall ing that F is a left U -mo du le, w e deﬁ ne an action of H on F by the form ula ( f ◮ ⊳ u ) g = f ( u ⊲ g ) , f , g ∈ F , u ∈ U . (3.34) Lemma 3.17. The ab ove formula deﬁnes an action, which makes F an H -mo dule c o algebr a. Pr o of. W e c hec k directly the action axiom ( f 1 ◮ ⊳ u 1 )  ( f 2 ◮ ⊳ u 1 ) g  = ( f 1 ◮ ⊳ u 1 )( f 2 u 2 ⊲ g ) = f 1 u 1 ⊲ ( f 2 u 1 ⊲ g ) = f 1 ( u 1 (1) ⊲ f 2 )( u 1 (2) u 2 ⊲ g ) =  f 1 u 1 (1) ⊲ f 2 ◮ ⊳ u 1 (2) u 2  g =  ( f 1 ◮ ⊳ u 1 )(( f 2 ◮ ⊳ u 1 )  g , 78 and the p rop erty of b eing a Hopf action: ∆(( f ◮ ⊳ u ) g ) = ∆( f u ⊲ g ) = f (1) u (1) < 0 > g (1) ⊗ f (2) u (1) < 1 > ( u (2) ⊲ g (2) ) = ( f (1) ◮ ⊳ u (1) < 0 > ) g (1) ⊗ ( f (2) u (1) < 1 > ◮ ⊳ u (2) ) g (2) . T o realize U as an H -mod ule and como d u le coal gebra, we identify it with C := H ⊗ F C , via the map ♮ : H ⊗ F C → U deﬁ n ed b y ♮ ( f ◮ ⊳ u ⊗ F 1) = ǫ ( f ) u. (3.35) Lemma 3.18 . The map ♮ : C → U is a n isomorphism of c o algebr as. Pr o of. The map is w ell-deﬁned, b ecause ♮ (( f ◮ ⊳ u )( g ◮ ⊳ 1) ⊗ F 1) = ♮ ( f u (1) ⊲ g ◮ ⊳ u (2) ⊗ F 1) = ǫ ( f u (1) ⊲ g ) u (2) = ǫ ( f ) ǫ ( g ) u = ♮ ( f ◮ ⊳ u ⊗ F ǫ ( g )) . Let us c hec k that ♮ − 1 ( u ) = (1 ◮ ⊳ u ) ⊗ F 1 is its in v erse. It is easy to see that ♮ ◦ ♮ − 1 = Id U . On th e other hand , ♮ − 1 ◦ ♮ ( f ◮ ⊳ u ⊗ F 1) = ♮ − 1 ( ǫ ( f ) u ) = (1 ◮ ⊳ u ) ⊗ F ǫ ( f ) = (1 ◮ ⊳ u (2) ) ⊗ F ǫ ( S − 1 ( u (1) ) ǫ ( f ) =  (1 ◮ ⊳ u (2) )( S − 1 ( u (1) ) ⊲ f ◮ ⊳ 1)  ⊗ F 1 = f ◮ ⊳ u ⊗ F 1 . Finally one chec ks the com ultiplicati vit y as follo ws: ♮ ⊗ ♮ (∆ C ( f ◮ ⊳ u ⊗ F 1)) = ♮ ( f (1) ◮ ⊳ u (1) ⊗ F 1) ⊗ ♮ ( u (2) < − 1 > f (2) ◮ ⊳ u (2) < 0 > ⊗ F 1) = ǫ ( f (2) ) u (1) ⊗ ǫ ( u (2) < − 1 > f (2) ) u (2) < 0 > = ǫ ( f (1) ) u (1) ⊗ ǫ ( f (2) ) u (2) = ∆ U ( ♮ ( f ◮ ⊳ u ⊗ F 1)) . By Prop osition 3.14, the follo wing is a map of H -mo d u le coa lgebras. Φ : H → C ◮ < F , Φ( h ) = ˙ h (1) ◮ < h (2) 1 . 79 T o b e able to use the Theorem 3.15 , we need to sho w that Φ is b ijectiv e. One has Φ( f ◮ ⊳ u ) = ˙ z }| { ( f (1) ◮ ⊳ u < 0 > ) ⊗ f (2) u < 1 > = ˙ z }| { (1 ◮ ⊳ ( u < 0 > (1) ))( S ( u < 0 > (2) ) ⊲ f (1) ◮ ⊳ 1)) ⊗ f (2) u < 1 > = ˙ z }| { 1 ◮ ⊳ u < 0 > ⊗ f u < 1 > ≡ u < 0 > ⊗ f u < 1 > , from which it follo ws th at Φ − 1 ( u ⊗ f ) = f S − 1 ( u < 1 > ) ◮ ⊳ u < 0 > . W e are now in a p osition to apply Theorem 3.15 to get a bico cyclic mo du le for computing the Hopf cyclic cohomo logy of H . W e would like, b efore that, to und erstand the action and coaction of H on C . O n e has ( f ◮ ⊳ u ) · ˙ z }| { ( g ◮ ⊳ v ) = ˙ z }| { f u (1) ⊲ g ◮ ⊳ u (2) v ≡ ε ( f ) ε ( g ) uv , whic h coincides with the natural action of H on U . Next, we tak e up the coaction of C . Recall the coac tion of H on C , H : C → H ⊗ C , H ( ˙ z }| { 1 ◮ ⊳ u ) = (1 ◮ ⊳ u ) (1) S (1 ◮ ⊳ u ) (3) ⊗ ˙ z }| { (1 ◮ ⊳ u ) (2) . Prop osition 3.19. The ab ove c o a ction c oincides on the original c o actio n of F on U . Pr o of. Using th e f act that U is co comm utativ e, one has (1 ◮ ⊳ u ) (1) S ((1 ◮ ⊳ u ) (3) ) ⊗ ˙ z }| { (1 ◮ ⊳ u ) (2) = (1 ◮ ⊳ u (1) < 0 > ) S ( u (1) < 2 > u (2) < 1 > ◮ ⊳ u (3) ) ⊗ ˙ z }| { u (1) < 1 > ◮ ⊳ u (2) < 0 > = (1 ◮ ⊳ u (1) < 0 > )(1 ◮ ⊳ S ( u (3) < 0 > )( S ( u (1) < 2 > u (2) < 1 > u (3) < 1 > ) ◮ ⊳ 1) ⊗ ⊗ ˙ z }| { u (1) < 1 > ◮ ⊳ u (2) < 0 > = (1 ◮ ⊳ u (1) < 0 > S ( u (3) < 0 > ))( S ( u (1) < 2 > u (2) < 1 > u (3) < 1 > ) ◮ ⊳ 1) ⊗ ˙ z }| { u (1) < 1 > ◮ ⊳ u (2) < 0 > = 80 whic h means that after id en tifying C with U one h as (1 ◮ ⊳ u (1) < 0 > S ( u (3) < 0 > ))( S ( u (1) < 1 > u (2) < 1 > u (3) < 1 > ) ◮ ⊳ 1) ⊗ u (2) < 0 > = (1 ◮ ⊳ u < 0 > (1) S ( u < 0 > (3) ))( S ( u < 1 > ) ◮ ⊳ 1) ⊗ u < 0 > (2) = 1 ◮ ⊳ S ( u < 1 > ) ⊗ u < 0 > ≡ S ( u < 1 > ) ⊗ u < 0 > . With the a b o v e iden tiﬁcations of a ctions and coacti ons, one has the follo wing bicyclic mo d ule Z ( H , F ; C δ ), where δ is the mo dular c haracter: . . . B U   . . . B U   . . . B U   C δ ⊗ U ⊗ 2 b F / / b U O O B U   C δ ⊗ F ⊗ U ⊗ 2 b F / / B F o o b U O O B U   C δ ⊗ F ⊗ 2 ⊗ U ⊗ 2 b F / / B F o o b U O O B U   . . . B F o o C δ ⊗ U b F / / b U O O B U   C δ ⊗ F ⊗ U b F / / B F o o b U O O B U   C δ ⊗ F ⊗ 2 ⊗ U b F / / B F o o b U O O B U   . . . B F o o C δ ⊗ C b F / / b U O O C δ ⊗ F ⊗ C b F / / B F o o b U O O C δ ⊗ F ⊗ 2 ⊗ C b F / / B F o o b U O O . . . B F o o , (3.36) A t this stage, w e introdu ce the follo wing Chev alley-Eilen b erg-t yp e bicom- plex: . . . ∂ g   . . . ∂ g   . . . ∂ g   C δ ⊗ ∧ 2 g β F / / ∂ g   C δ ⊗ F ⊗ ∧ 2 g β F / / ∂ g   C δ ⊗ F ⊗ 2 ⊗ ∧ 2 g β F / / ∂ g   . . . C δ ⊗ g β F / / ∂ g   C δ ⊗ F ⊗ g β F / / ∂ g   C δ ⊗ F ⊗ 2 ⊗ g ∂ g   β F / / . . . C δ ⊗ C β F / / C δ ⊗ F ⊗ C β F / / C δ ⊗ F ⊗ 2 ⊗ C β F / / . . . , (3.37) 81 Here ∂ g is th e Lie alg ebra homology b ound ary of g with coeﬃcients in right mo dule C δ ⊗ F ⊗ p ; explicitly ∂ g ( 1 ⊗ ˜ f ⊗ X 0 ∧ · · · ∧ X q − 1 ) = X i ( − 1) i ( 1 ⊗ ˜ f ) ⊳ X i ⊗ X 0 ∧ · · · ∧ c X i ∧ · · · ∧ X q − 1 + ( − 1) i + j X i . . . X q < 1 > ) ⊗ X 1 < 0 > ∧ · · · ∧ X q < 0 > In other w ords, β F is just the coalge br a cohomology cob oun d ary of th e coalge bra F with coeﬃcien ts in ∧ q g induced from the coactio n of F on U ⊗ q . H ∧ g : C δ ⊗ ∧ q g → F ⊗ C δ ⊗ ∧ q g , (3.39) H ( 1 ⊗ X 1 ∧ · · · ∧ X q ) = S ( X 1 < 1 > . . . X q < 1 > ) ⊗ 1 ⊗ X 1 < 0 > ∧ · · · ∧ X q < 0 > One notes that, since F is commutat ive , the coact ion H ∧ g is well -deﬁn ed. Prop osition 3.20. Th e b ic omp lexes (3.36) and (3.37) have quasi-isomorphic total c om plexes. Pr o of. W e apply antisymmetriza tion map ˜ α : C δ ⊗ F ⊗ p ⊗ ∧ q g → C δ ⊗ F ⊗ p ⊗ U ⊗ q ˜ α = Id ⊗ α, where α is the usual antisymmetriza tion map α ( X 1 ∧ · · · ∧ X p ) = 1 p ! X σ ∈ S p ( − 1) σ X σ (1) ⊗ . . . ⊗ X σ ( p ) . (3.4 0) 82 Its left in v erse is ˜ µ := Id ⊗ µ wher e µ is the natural left in v erse to α (see e.g. [19, p age 436]). As in the pro of of Prop osition 7 of [6], one has the follo wing comm utativ e diagram: C δ ⊗ F ⊗ p ⊗ ∧ q g ˜ α / / 0   C δ ⊗ F ⊗ p ⊗ U ⊗ q b U   C δ ⊗ F ⊗ p ⊗ ∧ q +1 g ˜ α / / ∂ g O O C δ ⊗ F ⊗ p ⊗ U ⊗ q +1 . B U O O Since ˜ α do es not aﬀect F ⊗ p , it is easy to see th at the follo wing diagram also comm utes: C δ ⊗ F ⊗ p ⊗ ∧ q g ˜ α / / β F   C δ ⊗ F ⊗ p ⊗ U ⊗ q b F   C δ ⊗ F ⊗ p +1 ⊗ ∧ q g ˜ α / / B F O O C δ ⊗ F ⊗ p +1 ⊗ U ⊗ q . B F O O Here B F is the Connes b oun d ary op erator for the cocyclic modu le C n F ( F , C δ ⊗ ∧ q g ), where F coacts on C δ ⊗ ∧ q g via (3.39) and acts trivially . Since F is comm utativ e and its actio n on C δ ⊗ ∧ q g is trivial, Theorem 3.22 of [22] implies that B F ≡ 0 in Hochsc hild c ohomology . On the other hand Theorem 7 [6] (whic h is applied here for co eﬃcien ts in a general mo dule), pro ve s that the columns of (3.36) with cob oundary B U and th e columns of (3.37) with coboun dary ∂ g are quasi-isomorphic. This ﬁn ishes the pr o of. 3.4 Applications 3.4.1 Hopf cyclic Chern classes In this section w e compu te the relativ e Hopf cyclic co homology of H n mo dulo the sub algebra L := U ( gl n ) with co eﬃcients in C δ . T o this end, we ﬁrst form the Hopf sub algebra K := L ◮ ⊳ F ⊂ H . Here L acts on F via its action inherited from L ⊂ U on F , and F coacts on L trivially . Th e second coalg ebra is C := H ⊗ K C . Letting h := gl n , and S := S ( g / h ) b e the symmetric algebra of the v ector sp ace g / h , we ident ify C with the coalgebra S , as follo ws. Since g ∼ = V > ⊳ h , where V = R n , w e can regard U as b eing U ( V ) > ⊳ U ( h ). The identiﬁca tion of C with S is ac hiev ed b y the map ( X > ⊳ Y ) ◮ ⊳ f ⊗ K 1 7→ ǫ ( Y ) ǫ ( f ) X, X ∈ U ( V ) , Y ∈ U ( h ) , f ∈ F . 83 As in Lemma 3.18, one c hec ks that this identiﬁca tion is a coalgebra iso- morphism. Finally , since g / h ∼ = V as v ector spaces, U ( V ) ∼ = S ( g / h ) as coalge bras. Similarly , one identiﬁes, K L := K ⊗ L C with F as coalgebras, via Y ◮ ⊳ f ⊗ L 1 7→ ǫ ( Y ) f , Y ∈ U ( h ) , f ∈ F . Let u s r ecall th e mixed complex ( C n , b, B ), wh ere C n := C n ( H , L ; C δ ) = C δ ⊗ L C ⊗ n , whic h computes relativ e Hopf cyc lic cohomology o f H mo d ulo L with co eﬃcien ts in C δ . W e also r ecall that the isomorphism Θ deﬁned in (3.32) id entiﬁes this complex with the diagonal complex Z ∗ , ∗ ( H , K , L ; C δ ) = C δ ⊗ L F ⊗ p ⊗ S ⊗ q . Now b y using Th eorem 3.15 w e get the follo w ing bicom- plex whose total co mplex is quasi isomorphic to C n via the Eilenberg-Zilb er Theorem. . . . B S   . . . B S   . . . B S   C δ ⊗ L S ⊗ 2 b F / / b S O O B S   C δ ⊗ L F ⊗ S ⊗ 2 b F / / B F o o b S O O B S   C δ ⊗ L F ⊗ 2 ⊗ S ⊗ 2 b F / / B F o o b S O O B S   . . . B F o o C δ ⊗ L S b F / / b S O O B S   C δ ⊗ L F ⊗ S b F / / B F o o b S O O B S   C δ ⊗ L F ⊗ 2 ⊗ S b F / / B F o o b S O O B S   . . . B F o o C δ ⊗ L C b F / / b S O O C δ ⊗ L F ⊗ C b F / / B F o o b S O O C δ ⊗ L F ⊗ 2 ⊗ C b F / / B F o o b S O O . . . B F o o , (3.41) Similar to (3.37 ), we in tro duce the follo wing bicomplex. 84 . . . ∂ ( g , h )   . . . ∂ ( g , h )   . . . ∂ ( g , h )   C δ ⊗ ∧ 2 V β F / / ∂ ( g , h )   C δ ⊗ F ⊗ ∧ 2 V β F / / ∂ ( g , h )   C δ ⊗ F ⊗ 2 ⊗ ∧ 2 V β F / / ∂ ( g , h )   . . . C δ ⊗ V β F / / ∂ ( g , h )   C δ ⊗ F ⊗ V β F / / ∂ ( g , h )   C δ ⊗ F ⊗ 2 ⊗ V ∂ ( g , h )   β F / / . . . C δ ⊗ C β F / / C δ ⊗ F ⊗ C β F / / C δ ⊗ F ⊗ 2 ⊗ C β F / / . . . , (3.42) Here V stands for the v ector space g / h and ∂ ( g , h ) is the relativ e Lie algebra homology b oundary of g relativ e to h with co eﬃcien ts in C δ ⊗ F ⊗ p with the action deﬁned in (3 .38). Th e cob oundary ˜ β F is induced by β F the coalg ebra cohomology cob oundary of F with trivial coeﬃcients in C δ ⊗ ∧ V . One notes that the coacti on of F on S is indu ced from the coaction of F on U via the natural pr o jection π : U → U ⊗ L C ∼ = S . Prop osition 3.21. Th e total c omplexes of the bic omplexes (3.41) and (3.42) ar e quasi isomorphic. Pr o of. The p r o of is similar to that of Prop osition 3.20 b ut more delicate. One replaces the anitsymmetrization map b y its relativ e version and instead of Prop osition 7 [6] one uses Theorem 15 [9 ]. On the other hand since L ⊂ H n is cocommutat ive , every map in the bicomplex (3.41) is L -linear, including th e h omotop y maps used in Eilenberg-Zilb er theorem [20], and in [19, pp. 438-442] , as w ell as in the sp ectral sequence u sed in [22, Theorem 3.22]. T o ﬁnd the cohomology of the ab ov e b icomplex w e ﬁ rst compu te the sub - complex consisting of gl n -in v arian ts. Deﬁnition 3.22. We say a map F ⊗ m → F ⊗ p is an or der e d pr oje ction of or der ( i 1 , . . . , i q ) , including ∅ or der, if i t of the form f 1 ⊗ . . . ⊗ f n 7→ f 1 . . . f i 1 − 1 ⊗ f i 1 . . . f i 2 − 1 ⊗ . . . ⊗ f i p . . . f m . (3.43) We denote the set of al l such pr oje ctions by Π m q . 85 Let S m denote the s ymmetric group of order m . Let also ﬁx σ ∈ S n − q , and π ∈ Π n − q p , wher e 0 ≤ p, q ≤ n . W e d eﬁne θ ( σ, π ) = (3.44) X ( − 1) µ 1 ⊗ π ( η j 1 µ (1) ,j σ (1) ⊗ . . . ⊗ η j n − q µ ( n − q ) ,j σ ( n − q ) ) ⊗ X µ ( n − q + 1) ∧ · · · ∧ X µ ( n ) where the s ummation is o v er all µ ∈ S n and all 1 ≤ j 1 , j 2 , . . . j q ≤ n . Lemma 3.23. The elements θ ( σ , π ) ∈ C δ ⊗ F ⊗ q ⊗ ∧ p V , with σ ∈ S n − p and π ∈ Π n − p q , ar e h -invariant and sp an the sp ac e C δ ⊗ h F ⊗ q ⊗ ∧ p V . Pr o of. W e iden tify C δ with ∧ n V ∗ as h -mo dules, by send in g 1 ∈ C δ in to the v olume elemen t ξ 1 ∧ . . . ∧ ξ n ∈ ∧ n V ∗ , wh ere { ξ 1 , . . . , ξ n } is the du al basis to { X 1 , . . . , X n } . On e obtains thus an isomorphism C δ ⊗ h F ⊗ p ⊗ ∧ q V ∼ =  ∧ n V ∗ ⊗ F ⊗ p ⊗ ∧ q V  h . (3.45) W e no w observ e that the relations (2.33) and (1.24) show that the ac tion of h on the jet co ord inates η i j j 1 ,...,j k s is tensorial. In particular, the cen tral elemen t of h , Z = P n i =1 Y i i , acts as a grading op erator, assigning degree 1 to eac h X ∈ V , degree − 1 to eac h ξ ∈ V ∗ and d egree k to η i j j 1 ,...,j k . As an immediate co nsequen ce, the space of in v arian ts (3.45) is see n to b e generated by monomials with the same num b er of up p er and lo we r indices, more pr ecisely  ∧ n V ∗ ⊗ F ⊗ p ⊗ ∧ q V  h =  ∧ n V ∗ ⊗ F ⊗ p [ n − q ] ⊗ ∧ q V  h , (3.46) where F ⊗ p [ n − q ] designates the homogeneous comp onen t of F ⊗ q of degree n − q . F ur th ermore, using (2.28), (2.2 9 ) we can th ink of F as generated by α i j j 1 ,...,j k s instead of the η i j 1 ,...,j k s. The adv antag e is th at the α i j 1 ,...,j k s are symmetric in all low er indices and freely g enerate the algebra F . Fixing a lexicographic ordering on the set of m ulti-indices i j j 1 ,...,j k , one ob tains a PBW-basis of F , view ed as th e Lie algebra generated by the α i j j 1 ,...,j k s. This allo ws us to extend the assignment α i j j 1 ,...,j k 7→ ξ i ⊗ X j ⊗ X j 1 ⊗ . . . ⊗ X j k ∈ V ∗ ⊗ V ⊗ k +1 , ﬁrst to the p ro ducts wh ic h deﬁn e the PBW-basis and next to an h -equiv ariant em b edd ing of F ⊗ q [ n − q ] in to a ﬁn ite direct sum of tensor pro ducts of the 86 form ( V ∗ ) ⊗ r ⊗ V ⊗ s . Sending the we dge pr o duct of v ectors into the an tisym- metrized tensor pro duct, w e thus obtain an embed ding  ∧ n V ∗ ⊗ F ⊗ p [ n − q ] ⊗ ∧ q V  h ֒ →  ( V ∗ ) ⊗ n ⊗ X r,s ( V ∗ ) ⊗ r ⊗ V ⊗ s ⊗ V ⊗ q  h . (3.47) In the righ t hand s ide we are no w dealing with the cla ssical theory of tensor in v arian ts for GL( n, R ), cf. [27]. All su c h in v arian ts are linear combinatio ns tensors of the form T σ = X ξ j 1 ⊗ . . . ⊗ ξ j n ⊗ X j σ (1) ⊗ . . . ⊗ X j σ ( p ) ⊗ . . . . . . ⊗ ξ j r ⊗ . . . ⊗ ξ j m ⊗ . . . ⊗ X j σ ( s ) ⊗ . . . ⊗ X j σ ( m ) , where σ ∈ S m and the s um is o v er all 1 ≤ j 1 , j 2 , . . . j m ≤ n . Because the antisymmetriza tion is a pr o jection o p erator, suc h linear com- binations b elong to the subspace in the left hand side of (3.47) only when they are totally an tisymmetric in the ﬁ rst n -co v ectors. Recalling the id en- tiﬁcation 1 ∼ = ξ 1 ∧ . . . ∧ ξ n , and the fact that α i j k = η i j k , while the higher α i j j 1 ,...,j k s are symmetric in all lo wer in dices, one easily sees that the pro- jected inv arian ts b elong to the linear span of those of the form (3.44 ). F or eac h partition λ = ( λ 1 ≥ . . . ≥ λ k ) of th e set { 1 , . . . , p } , wh ere 1 ≤ p ≤ n , w e let λ ∈ S p also den ote a permutation whose cycles ha v e lengths λ 1 ≥ . . . ≥ λ k , i.e. represen ting the corresp on d ing conjugacy class [ λ ] ∈ [ S p ]. W e then deﬁn e C p,λ := X ( − 1) µ 1 ⊗ η j 1 µ (1) ,j λ (1) ∧ · · · ∧ η j p µ ( p ) ,j λ ( p ) ⊗ X µ ( p +1) ∧ · · · ∧ X µ ( n ) , where the s ummation is o v er all µ ∈ S n and all 1 ≤ j 1 , j 2 , . . . , j p ≤ n . Theorem 3.24. The c o c hains { C p,λ ; 1 ≤ p ≤ n, [ λ ] ∈ [ S p ] } ar e c o cycles and their classes form a b a sis of the gr o up H P ǫ ( H n , U ( gl n ); C δ ) , wher e ǫ ≡ n mo d 2 . The c omplementa ry p arity gr oup H P 1 − ǫ ( H n , U ( gl n ); C δ ) = 0 . Pr o of. Let d b e the comm utativ e Lie algebra generated by all η i j,k , and let F = U ( d ), b e the p olynomial algebra of η i j,k . Obvio usly F is a Hopf subalgebra of F and stable under the act ion of h . By using Lemma 3.44, we ha v e: C δ ⊗ h F ⊗ p ⊗ ∧ q V ∼ = C δ ⊗ h F ⊗ p ⊗ ∧ q V . (3.48) Hence we reduce the problem to compute the cohomology of ( C d ⊗ h F ⊗∗ ⊗ ∧ q V , b F ), where β F is in d uced by the Hoc hschild cob ound ary of the coalgebra 87 F with trivial co eﬃcien ts. O ne uses the fact that ( C δ ⊗ F ⊗∗ ⊗ ∧ q V , b F ) and ( C δ V ⊗ ∧ ∗ d ⊗ ∧ q , 0) are homotop y equiv alen t and the homotop y is h -linear [19] to see that the q th c ohomology of t he complex under consid er ation is C δ ⊗ h ∧ p d ⊗ ∧ q V . S imilar to the proof of Lemma 3.44 w e replac e C δ with ∧ n V ∗ . W e also replace α i j,k with ˜ ξ i ⊗ X ′ j ⊗ X ′′ k , where ˜ ξ i , X ′ j and X ′′ k are basis for V ∗ , V and V r esp ectiv ely . Using the ab ov e iden tiﬁcation one has C δ ⊗ h ∧ p d ⊗ ∧ q V ∼ =  ∧ n V ∗ ⊗ ( ∧ p V ∗ ⊗ ∧ p V ⊗ ∧ p V ) ⊗ ∧ q V  h . Via the same argumen t as in th e pro of of Lemma 3. 44 , o ne co nclud es that the in v arian t space is generated b y elemen ts of the form C p,σ := X ( − 1) µ 1 ⊗ η j 1 µ (1) ,j σ (1) ∧ · · · ∧ η j p µ ( p ) ,j σ ( p ) ⊗ X µ ( p +1) ∧ · · · ∧ X µ ( n ) (3.49) where σ ∈ S p , and the su mmation is take n as in (3.44 ). No w let σ = σ 1 . . . σ k , where σ 1 = ( a, σ ( a ) , . . . , σ α − 1 ( a )), . . . , σ k = ( z , σ ( z ) , . . . , σ ζ − 1 ( z )) . Since distinct cyclic p erm utations comm ute among eac h other we may assume α ≥ β · · · ≥ γ ≥ ζ . W e deﬁn ed the follo wing tw o p er mutations. τ := (1 , 2 , . . . , α )( α + 1 , . . . , β ) , . . . , ( α + · · · + γ + 1 , . . . , α + · · · + ζ ) , θ ( i ) =              σ i − 1 ( a ) , 1 ≤ i ≤ α σ i − 1 − α ( b ) , 1 + α ≤ i ≤ α + β . . . . . . σ i − 1 − α −···− γ ( z ) , α + · · · + γ + 1 ≤ i ≤ p i, p + 1 ≤ i ≤ n W e claim C p,σ and C p,τ coincide up to a sign. Ind eed, C p,σ = ± X µ ( − 1) µ η j a µ ( a ) ,j σ ( a ) ∧ η j σ ( a ) µ ( σ ( a )) ,j σ 2 ( a ) ∧ . . . ∧ η j σ α − 1 ( a ) µ ( σ α − 1 ( a )) ,j a ∧ η j b µ ( b ) ,j σ ( b ) ∧ η j σ ( b ) µ ( σ ( b )) ,j σ 2 ( b ) ∧ . . . ∧ η j σ β − 1 ( b ) µ ( σ β − 1 ( b )) ,j b ∧ · · · ∧ η j z µ ( z ) ,j σ ( z ) ∧ η j σ ( z ) µ ( σ ( z )) ,j σ 2 ( z ) ∧ . . . ∧ η j σ ζ − 1 ( z ) µ ( σ ζ − 1 ( z )) ,j z ⊗ X ( µ ) = ± X µ ( − 1) µ η j 1 µ ( a ) ,j 2 ∧ η j 2 µ ( σ ( a )) ,j 3 ∧ . . . ∧ η j α µ ( σ α ( a )) ,j 1 ∧ η j α +1 µ ( b ) ,j α +2 ∧ η j α +2 µ ( σ ( b )) ,j α +3 ∧ . . . ∧ η j α + β µ ( σ β ( b )) ,j α +1 ∧ · · · ∧ η j α + ··· + γ +1 µ ( z ) ,j α + ··· + γ +2 ∧ η j α + ··· + γ +2 µ ( σ ( z )) ,j α + ...γ + 3 ∧ . . . ∧ η j α + ··· + ζ µ ( σ ζ ( z )) ,j α + ...γ +1 ⊗ X ( µ ) ⊗ = ± X µ ( − 1) µ η j 1 µθ (1) ,j τ (1) ∧ · · · ∧ η j q µθ ( p ) ,j τ ( q ) ⊗ X ( µθ ) = ± C p,τ . 88 Here X ( µ ) stands for X µ ( p +1) ∧ · · · ∧ X µ ( n ) . The fact that { C p,λ ; 1 ≤ p ≤ n, [ λ ] ∈ [ S p ] } are linearly indep endent , and therefore form a basis for C δ ⊗ h F ⊗ p ⊗ ∧ q V , f ollo ws from the observ ation that, if σ ∈ S p , the term X ( − 1) µ 1 ⊗ η 1 µ (1) ,σ (1) ∧ · · · ∧ η p µ ( p ) ,σ ( p ) ⊗ X µ ( p +1) ∧ · · · ∧ X µ ( n ) , app ears in C p,τ if and only if [ σ ] = [ τ ]. These are all the p eriod ic cyclic classes, b ecause all of them s it on the n th sk ew diagonal of (3.42) and there are n o other inv arian ts. Therefore B ≡ ∂ g , h = 0 on the inv ariant space. In particular dim H P ∗ ( H n , U ( gl n ); C δ ) = p (0) + p (1) + . . . + p ( n ) , where p denotes th e p artition fu n ction, whic h is the same as the dimen- sion of the truncated Ch ern ring P n [ c 1 , . . . c n ]. Moreo v er, as noted in the in tro du ction, the assignment C p,λ 7→ c p,λ := c λ 1 · · · c λ k , λ 1 + . . . + λ k = p ; deﬁnes a linear isomorphism b et w een H P ∗ ( H n , U ( gl n ); C δ ) and P n [ c 1 , . . . c n ]. 3.4.2 Non-p erio dized Hopf cyclic cohomo logy of H 1 In this section we apply the bicomplex (3.37) to compu te all cyclic coho- mology of H 1 with co eﬃcien ts in the MPI (1 , δ ). Ac tually , inv o king the equiv al ence (3.6), we shall compute H C ( H cop 1 , C δ ) instead. Let us recall th e present ation of the Hopf algebra H 1 . As an algebra, H 1 is generated by X , Y , δ k , k ∈ N , su b ject to the relations [ Y , X ] = X , [ Y , δ k ] = k δ k , [ X , δ k ] = δ k +1 , [ δ j , δ k ] = 0 . Its coalgebra str ucture is un iquely determined by ∆( Y ) = Y ⊗ 1 + 1 ⊗ Y , ∆( X ) = X ⊗ 1 + 1 ⊗ X + δ 1 ⊗ Y , ∆( δ 1 ) = δ 1 ⊗ 1 + 1 ⊗ δ 1 , ǫ ( X ) = ǫ ( Y ) = ǫ ( δ k ) = 0 . By Th eorem (2.15), one h as H cop 1 ∼ = F ◮ ⊳ U . W e next apply Theorem 3.15, which gives a qu asi-isomorphism b et w een C ∗ ( F ◮ ⊳ U , C δ ) and the 89 total complex of the bicomplex (3.37 ). One n otes that Y grades H 1 b y [ Y , X ] = X , [ Y , δ k ] = k δ k . Accordingly , Y grades U , and also F by Y ⊲ η 1 1 ,..., 1 = k η 1 1 ,... 1 , where k + 1 is the n umber of lo we r in dices. Hence Y grades the b icomplex (3.37). O n e notes that every identiﬁca tion, isomorphism , and homotopy whic h h as b een used to pass from the bicomplex C ∗ ( F ◮ ⊳ U , C δ ) to (3.37) resp ects this grading ( cf. also [24]). As a r esu lt, on e can r elativize the computations to eac h homogeneous comp onen t. Th e degree of ˜ f ∈ F ⊗ q will b e denoted by | ˜ f | . W e next r ecall Gonˇ caro v a’s results [14] co ncernin g the Lie algebra cohomol- ogy of n . T aking as b asis of n the v ector ﬁ elds e i = x i +1 d dx ∈ n , i ≥ 1 , and denoting the dual basis by { e i ∈ n ∗ | i ≥ 1 } , one iden tiﬁes ∧ k n ∗ with the totally antisymmetric p olynomials in v ariables z 1 . . . z k , via the m ap e i 1 ∧ · · · ∧ e i k 7→ X µ ∈ S k ( − 1) µ z i µ (1) 1 . . . z i µ ( k ) k . According to [14], for eac h dimension k ≥ 1, the Lie algebra cohomology group H k ( n , C ) is tw o-dimensional and is generated by the classes ω k := z 1 . . . z k Π 3 k , ω ′ k := z 2 1 . . . z 2 k Π 3 k , where Π k := Π 1 ≤ i

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