The Hochschild cohomology ring of the standard Podles quantum sphere

THE HOCHSCH ILD COHOMO LOGY RING O F T H E ST AND ARD PODLE ´ S Q UANT UM SPHERE ULRICH KR ¨ AHMER Abstra ct. The cup and cap pro duct in twisted Ho chsc hild (co)homolo- gy is computed for the standard qu antum 2-sphere and used to construct a cyclic 2-co cycle th at represen ts the fundamental Ho chschil d class. 1. I n troduction The aim of this a rticle is to co mpute for the quan tised co ord inate ring A = C q [ S 2 ] (the standard P od le ´ s quantum 2-sphere) t he cup and cap pro duct ` : H m ( A, σ A ) ⊗ H n ( A, τ A ) → H m + n ( A, τ ◦ σ A ) , a : H n ( A, σ A ) ⊗ H m ( A, τ A ) → H n − m ( A, τ ◦ σ A ) in the Ho c hsc hild (co)homolog y of A w ith co efficient s in th e b imo dules σ A that arise by t w isting the canonical bimo du le structur e of A b y σ ∈ Aut( A ). In [8] we carried out similar computations for C q [ S U (2)]. F or C q [ S 2 ] they b ecome m uc h simpler and their conceptual con ten t that w as somewhat hid- den in [8] b et ween lengthy computations b ecomes more transparent. F or the co ordin ate ring of a smo oth v ariet y X , the Ho c hsc hild-Kostan t- Rosen b erg theorem identifies Λ( A ) := ( L n ≥ 0 H n ( A, A ) , ` ) with the alge bra of m ultiv ector fields on X , and L n ≥ 0 H n ( A, A ) as a Λ( A )-mo dule (via a ) with the d ifferential forms Ω( X ) on X . F or n oncomm utativ e algebras, Λ( A ) tends to b e fairly degenerate. Ho wev er, t wisting b y σ allo ws one to consid er ric her cohomology rings that enco d e more inform ation ab out A . F or example, the generators of th e Drinfeld-Jim b o quanti sation of the Lie algebra of S U (2) act via t wisted rather th an usual deriv ations on A = C q [ S 2 ]. These give rise to tw o cohomology classes [ ∂ ± 1 ] ∈ H 1 ( A, σ − 1 mod A ), where σ mod is W oronowicz ’s mo du lar auto morph ism determined for example b y (2) b elo w. W e will see that they b eh a ve u nder the cup p ro duct similar to the corresp ondin g classical S U (2)-in v arian t vecto r fields on S 2 = S U (2) /U (1), [ ∂ 1 ] ` [ ∂ 1 ] = [ ∂ 1 ] ` [ ∂ − 1 ] + q 2 [ ∂ − 1 ] ` [ ∂ 1 ] = [ ∂ − 1 ] ` [ ∂ − 1 ] = 0 , and use them to define a functional on H 2 ( A, σ mod A ) ≃ C of the form (1) ϕ ([ ω ]) := q − 1 Z [ ω ] a ([ ∂ 1 ] ` [ ∂ − 1 ]) ∈ C . Here [ ω ] ∈ H 2 ( A, σ mod A ) is acted on by [ ∂ 1 ] ` [ ∂ − 1 ] ∈ H 2 ( A, σ − 2 mod A ) to pro du ce a class in H 0 ( A, σ − 1 mod A ), and then one applies a certain twisted trace R ∈ ( H 0 ( A, σ A )) ∗ in ord er to obtain a numerical inv ariant of [ ω ]. 1 2 ULRICH KR ¨ AHMER The fun ctional ϕ pr o v id es a dual descrip tion of the fun damen tal class [ d A ] ∈ H 2 ( A, σ mod A ) that corresp onds u n der the P oincar ´ e -t yp e dualit y [12] (2) H n ( A, σ mod A ) ≃ H 2 − n ( A, A ) to 1 ∈ H 0 ( A, A ). F rom th e practical p oin t of view, one can u se ϕ to deter- mine th e homology class of a giv en Ho chsc hild cycle, and for C q [ S U (2)] this to ol allo wed us to compute the cyclic h omology [13] built up on H n ( A, σ A ) as a sp ecial case of Connes-Mosco vici’s Hopf-cyclic homology [3]. The trace R in (1 ) is f or C q [ S 2 ] actually a c haracter (namely the restriction of the counit ε of C q [ S U (2)] to C q [ S 2 ]), s o on th e lev el of c hains a 0 ⊗ a 1 ⊗ a 2 in th e standard Ho chsc hild complex, ϕ acts as ϕ ( a 0 , a 1 , a 2 ) = q − 1 ε ( a 0 ) F ( a 1 ) E ( a 2 ) , where E , F : A → k are th e (u nt w isted) d eriv ations giv en b y E ( a ) := ε ( ∂ − 1 ( a )) , F ( a ) := ε ( ∂ 1 ( a )) . There seems to b e a general principle b ehin d this that we observ ed already in [8]. Therein, the trace R needed wa s the in tegral ov er the un q u an tised maximal torus in quan tum S U (2). F or th e quantum 2-sphere the corre- sp ond ing submanifold of S 2 is simply a p oint, namely one of the t w o lea v es of the symp lectic foliation of the Po isson manifold S 2 quan tised b y A . Finally we discus s ho w to add a count er term η to ϕ in order to ob- tain a f unctional on cyclic homology without c hanging the fun ctional on H 2 ( A, σ mod A ). Sc hm ¨ udgen and W agner h a v e defi ned a cyclic 2-cocycle in [17] that lo oks like ϕ , only the trace R is the Haar fun ctional of C q [ S U (2)], and this makes their fu n ctional trivial on Ho c hsc hild homology [6]. The structur e of the pap er is as follo ws. In Sections 2-5 we recall the def- inition of Ho chsc hild (co)homology , of the cup and cap pr o duct, giv e some more bac kground ab out the ab o ve m en tioned Poinca r´ e dualit y and introdu ce then the algebra A = C q [ S 2 ] we wa nt to study . In S ection 6 we recall f r om [6] an explicit formula for the fund amen tal class d A of A . In Section 7 we determine th e t w isted cen tre of A and its cap pro du ct action on th e second Ho c hsc hild h omology: in [6] it w as shown th at H 2 ( A, σ A ) = 0 except wh en σ = σ n mod for some n ≥ 1, and then H 2 ( A, σ n mod A ) ≃ C . Here we identify the sum of all these n on trivial homology groups with the f r ee mo dule of rank 1 o v er a p olynomial ring k [ x 0 ] that constitutes the t wisted cen tre of A . W e conti nue in S ection 8 with recalling from [6] (but in a slightl y simp lified form) th e compu tation of the zeroth Ho c hsc hild homology group s of A and of the t wisted traces on A , describing in addition the cap pro d uct action of the t wisted centre. Section 9 is the fi rst really int eresting one, here we compute the cup pro du ct b etw een t wisted deriv ations of A that arise from the ac tion of the qu an tised Lie algebra of S U (2) on A . W e observe similar as in [8 ] that these t wisted der iv ations generate a quanti sed exterior algebra. These compu tations are then used in Section 10 to defin e and d iscuss ϕ . Section 11 recalls the definition and some key pr op erties of cyclic homology , and finally w e show in Section 12 that ϕ can b e altered by a Ho c hs child cob oundary to obtain a cyclic co cycle. I ackno wledge sup p ort by the EPS R C fello wship EP/E/043267/1 . THE HO CHSCHILD COHOMOLOGY RING OF T H E ST ANDARD PODLE ´ S QUAN T U M SPH E R E 3 2. Ho chschild (co )homolo gy with coe f ficients in σ A Let A b e a unital asso ciativ e algebra ov er a field k and σ ∈ Aut( A ) b e an automorphism. W e denote by σ A the A -bimo du le whic h is A as vect or space w ith left and right A -actions giv en by a ⊲ b ⊳ c := σ ( a ) bc , a, b, c ∈ A , and b y H n ( A, σ A ) and H n ( A, σ A ) the Ho chsc hild (co)homology group s of A with co efficient s in σ A . Explicitly , H n ( A, σ A ) is the homology of the c hain complex C σ n := A ⊗ n +1 with b oundary map b n : C σ n → C σ n − 1 giv en b y b n ( a 0 ⊗ . . . ⊗ a n ) = n − 1 X i =0 ( − 1) i a 0 ⊗ . . . ⊗ a i a i +1 ⊗ . . . ⊗ a n +( − 1) n σ ( a n ) a 0 ⊗ . . . ⊗ a n − 1 . Usually w e will write b ( a 0 , . . . , a n ) instead of b n ( a 0 ⊗ . . . ⊗ a n ) and similarly for other multilinear maps. Dually , H n ( A, σ A ) is the cohomology of the co c hain complex C n σ := Hom k ( A ⊗ n , A ) with cob oundary map give n b y ( b n ψ )( a 0 , . . . , a n ) = σ ( a 0 ) ψ ( a 1 , . . . , a n ) + n − 1 X i =0 ( − 1) i +1 ψ ( a 0 , . . . , a i a i +1 , . . . , a n ) +( − 1) n +1 ψ ( a 0 , . . . , a n − 1 ) a n . In degree 0, we identi fy ψ : A ⊗ 0 := k → A with a := ψ (1) ∈ A . This is a co cycle precisely when ab = σ ( b ) a for all b ∈ A . Thus H 0 ( A, σ A ) consists of the σ -cent ral elemen ts of A . In degree 1, a co cycle is a σ -t wisted deriv ation ψ : A → A , ψ ( ab ) = σ ( a ) ψ ( b ) + ψ ( a ) b , and H 1 ( A, σ A ) is the space of all suc h deriv ations mo d ulo those of the form ψ ( a ) = ba − σ ( a ) b for some b ∈ A . F or more information and details, see e.g. [2, 7, 13, 14, 20]. 3. The cup a n d cap product The cup pro duct is the map ` : H m ( A, σ A ) ⊗ H n ( A, τ A ) → H m + n ( A, τ ◦ σ A ) , σ, τ ∈ Aut( A ) giv en on the leve l of co chains by (3) ( ϕ ` ψ )( a 1 , . . . , a m + n ) = τ ( ϕ ( a 1 , . . . , a m )) ψ ( a m +1 , . . . , a m + n ) . F or an y monoid G ⊂ Aut( A ), it tu rns Λ G ( A ) := M n ∈ N ,σ ∈ G H n ( A, σ A ) in to an N × G -graded algebra that we would like to view as s ome analogue of an algebra of multiv ector fields on a classical space. Twisted d eriv ations pla y here the r ole of v ector fi elds, and the follo wing easily c hec ked (see [8 ]) relations d emonstrate th eir b ehavio ur und er ` : lemma 3.1 . In de gr e e 0, ` r e duc es to the opp osite pr o duct of A , (4) a ` b = ba, a ∈ H 0 ( A, σ A ) , b ∈ H 0 ( A, τ A ) , and for c ∈ H 0 ( A, σ A ) and twiste d derivations ϕ ∈ C 1 σ , ψ ∈ C 1 τ we have ψ ` c = τ ( c ) ` ψ , [ ϕ ] ` [ ψ ] = − [ σ − 1 ◦ ψ ◦ σ ] ` [ ϕ ] ∈ H 2 ( A, τ ◦ σ A ) . 4 ULRICH KR ¨ AHMER Dually , Ω G ( A ) := M n ∈ N ,σ ∈ G H n ( A, σ − 1 A ) b ecomes an N × G -graded (right) mo du le o v er Λ G ( A ) via the cap pro d uct a : H n ( A, σ A ) ⊗ H m ( A, τ A ) → H n − m ( A, τ ◦ σ A ) , m ≤ n. Explicitly , this is giv en b etw een ϕ ∈ C m τ and a 0 ⊗ . . . ⊗ a n ∈ C σ n b y ( a 0 ⊗ . . . ⊗ a n ) a ϕ = τ ( a 0 ) ϕ ( a 1 , . . . , a m ) ⊗ a m +1 ⊗ . . . ⊗ a n ∈ C τ ◦ σ n − m . In particular, the cap pro du ct w ith a twisted cen tral elemen t c ∈ H 0 ( A, σ A ) is simp ly give n b y multiplicatio n from the left , (5) ( a 0 ⊗ . . . ⊗ a n ) a c = σ ( a 0 ) c ⊗ . . . ⊗ a n = ca 0 ⊗ . . . ⊗ a n . F or more information, see e.g. [2 , 8, 15]. 4. Poincar ´ e duality The cup and cap pr o duct stru ctures are inti mately related to P oincar ´ e- t yp e dualitie s b et w een homolog y and cohomology . As b ecame clear in recen t y ears, there is for many algebras A a d istinguished automorph ism σ mod and for all τ ∈ Aut( A ) a canonical k -linear isomorph ism (6) H n ( A, τ A ) ≃ H dim( A ) − n ( A, τ ◦ σ mod A ) , where dim( A ) is the dimension of A in the sense of [2], see e.g. [1, 5, 7, 11, 12] but fir st of all [19] for this story . Und er the ab o v e isomorphism, th e canon- ical elemen t 1 ∈ H 0 ( A, A ) corresp onds to a class [ d A ] ∈ H dim( A ) ( A, σ mod A ), and then the isomorphism is giv en by th e ca p pro duct w ith this fund amen tal class. In [8] we carried this out explicitly f or the standard quan tised coord i- nate r ing C q [ S U (2)] (see e.g. [10] for bac kground on quantum groups), and the aim h ere is to do the same for the standard quant um 2-sphere of P o dle ´ s that we in tro duce in the next section. F or the co ordinate ring of a smo oth affine v ariet y suc h a dualit y will hold if and only if th e v ariety is Cala bi-Y au, that is, if the line bun dle on X whose sections are the top degree K¨ ahler differen tials Ω dim( X ) ( X ) is trivial (in general one h as to t wist not by an au- tomorphism but by this mo dule, see e.g. [11]). This happ ens if and only if there is a no where (i.e. in no lo calisation at pr ime ideals) v anishing elemen t in Ω dim( X ) ( X ), and under the Ho c hsc hild-Kostan t-Rosen b erg isomorphism Ω dim( X ) ( X ) ≃ H dim( X ) ( k [ X ] , k [ X ]) su c h an elemen t w ill b e ident ified with the fu ndamenta l class [ d k [ X ]]. 5. The Podle ´ s sphere F rom no w on we fix k = C , an element q ∈ k \ { 0 } assumed to b e not a ro ot of unit y , and A is the stand ard Podle ´ s quantum 2-sph ere [16], that is, the un iv ersal k -algebra generated b y x − 1 , x 0 , x 1 satisfying the relations x ± 1 x 0 = q ∓ 2 x 0 x ± 1 , x ± 1 x ∓ 1 = q ∓ 2 x 2 0 + q ∓ 1 x 0 . It follo ws easily from these relations that the elemen ts e ij :=  x i 0 x j 1 j ≥ 0 , x i 0 x − j − 1 j < 0 , i ∈ N , j ∈ Z THE HO CHSCHILD COHOMOLOGY RING OF T H E ST ANDARD PODLE ´ S QUAN T U M SPH E R E 5 form a vecto r s pace basis of A . W e denote b y G the automorphism group of A . The defin in g relations imply that for any λ ∈ k \ { 0 } there is a un iqu e σ λ ∈ G with σ λ ( x n ) = λ n x n . lemma 5.1 . A ny σ ∈ G is of the form σ λ for some λ , that is, G ≃ k \ { 0 } . Pro of. It is str aightforw ard to classify the characte rs of A and to see that the in tersection of their k ernels is the ideal generated b y x 0 . It fol- lo w s that an y automorphism σ maps x 0 to a nonzero scalar multiple of x 0 . Hence x 0 σ ( x ± 1 ) = q ± 2 σ ( x ± 1 ) x 0 whic h implies σ ( x ± 1 ) = f ± x ± 1 for some f ± ∈ k [ x 0 ]. Inserting in to the defining relations gives the claim. ✷ See also [4, 10] for m ore information ab out this algebra. 6. The fundament al class As sho wn in [12], A satisfies (6) with d im( A ) = 2 (as it probably s hould b e for a qu an tum 2-sphere), and with σ mod determined uniquely b y σ mod ( x n ) = q 2 n x n . So H 2 ( A, σ mod A ) ≃ H 0 ( A, A ), the cen tre of A . This consists only of the scalars, hence [ d A ] is u nique up to n ormalisation. Hadfi eld has computed explicit vect or space bases of all H n ( A, σ A ) for general automorphisms σ [6] and has giv en in particular an explicit cycle representing [ d A ] ∈ H 2 ( A, σ mod A ): d A := 2 x 1 ⊗ ( x − 1 ⊗ x 0 − q 2 x 0 ⊗ x − 1 ) +2 x − 1 ⊗ ( q − 2 x 0 ⊗ x 1 − x 1 ⊗ x 0 ) +1 ⊗ ( q x 1 ⊗ x − 1 − q − 1 x − 1 ⊗ x 1 + ( q − q − 1 ) x 0 ⊗ x 0 ) +2 x 0 ⊗ ( x 1 ⊗ x − 1 − x − 1 ⊗ x 1 + ( q 2 − q − 2 ) x 0 ⊗ x 0 ) . 7. The twiste d centre and H 2 ( A, σ A ) As w e p oint ed out ab o ve, the fundamen tal class is classica lly (meaning for the co ordinate ring of a smo oth v ariet y ) represente d by a no w here v anishin g algebraic differen tial form of top degree, and this form can b e m ultiplied by an y r egular fun ction to giv e a new top degree f orm. Analogously we can act via th e cap pr o duct by an y t wisted cen tral elemen t a ∈ H 0 ( A, τ A ) on the fundamental class [ d A ] ∈ H 2 ( A, σ mod A ) and obtain another homology class in H 2 ( A, τ ◦ σ mod A ). This op eration clarifies completely the structure of all the other non v anishing H 2 ( A, σ A ) that w ere computed b y Hadfield, since they b ecome altogether identified with a free k [ x 0 ]-mo dule of rank 1: lemma 7.1 . The twiste d c entr e Λ 0 G ( A ) of A is the sub algebr a gener ate d by x 0 ∈ H 0 ( A, σ mod A ) , and f or every [ ω ] ∈ H 2 ( A, σ A ) , σ any automorp hism, ther e exists exactly one p olynomial f ∈ k [ x 0 ] such that [ ω ] = [ d A ] a f . Pro of. Since an y automorph ism fixes x 0 , an y twiste d central elemen t m ust commute with x 0 and is hence a p olynomial in x 0 (use the ve ctor space basis e ij ). C on v ersely , it is clear that x 0 ∈ H 0 ( A, σ mod A ) and hence x i 0 = x 0 ` . . . ` x 0 ∈ H 0 ( A, σ q 2 i A ) (recall Lemm a 3.1). The second p art follo ws b y P oincar ´ e d ualit y (6), but of cours e also from Hadfield’s explicit computations of all the nontrivia l H 2 ( A, σ A ). ✷ 6 ULRICH KR ¨ AHMER 8. Twiste d traces and H 0 ( A, σ A ) As a vec tor sp ace, H 0 ( A, σ A ) can b e describ ed as follo ws [6]: lemma 8.1 . F or σ = σ λ , the fol lowing is a ve ctor sp ac e b asis of H 0 ( A, σ A ) : { [1] } ∪ { [ x j ± 1 ] | j 6 = 0 , λ = 1 } ∪ { [ x 0 ] | λ 6 = q 2 i , i > 0 } ∪ { [ x i 0 ] | λ = q 2 i , i > 0 } . Pro of. By defin ition, H 0 ( A, σ A ) is as a vec tor space the qu otien t of A b y the subspace spann ed b y elemen ts of the form b ( a, b ) = ab − σ ( b ) a . Since a ⊗ bc = ab ⊗ c + σ ( c ) a ⊗ b − b ( a, b, c ) , one has b ( a, bc ) = b ( ab, c ) + b ( σ ( c ) a, b ), so im b is spanned by the elements b k ij := b ( e ij , x k ), i ∈ N , j ∈ Z , k = − 1 , 0 , 1 wh ic h are for σ = σ λ giv en by b − 1 ij = (1 − λ − 1 q 2 i ) e ij − 1 , j ≤ 0 b − 1 ij = ( q − 4 j +2 − λ − 1 q 2 i +2 ) e i +2 j − 1 + ( q − 2 j +1 − λ − 1 q 2 i +1 ) e i +1 j − 1 , j > 0 , b 0 ij = ( q − 2 j − 1) e i +1 j , b 1 ij = (1 − λq − 2 i ) e ij +1 , j ≥ 0 b 1 ij = ( q − 4 j − 2 − λq − 2 i − 2 ) e i +2 j +1 + ( q − 2 j − 1 − λq − 2 i − 1 ) e i +1 j +1 , j < 0 . Reducing this b y sheer insp ection giv es that im b is spanned by the elemen ts e i +1 j , ( λ − 1) e 0 j , j 6 = 0 , ( λ − q 2 i +4 ) q − 2 i − 2 e i +20 + ( λ − q 2 i +2 ) q − 2 i − 1 e i +10 , i ≥ 0 . The claim follo ws easily . ✷ Dually , H 0 ( A, σ A ) can b e d escrib ed in terms of σ -t wisted traces, that is, linear functionals R : A → k satisfying Z ab = Z σ ( b ) a, a, b ∈ A. Suc h traces ob viously descend to w ell-defined fu nctionals on H 0 ( A, σ A ) wh ich w e den ote for simplicit y b y the same sym b ol. F or eac h of the basis elements in L emm a 8.1, we can (and do) defi ne one su c h trace Z [ x j ± 1 ] e k l :=  1 k = 0 , ± j = l , 0 otherwise , j ≥ 0 , Z [ x 0 ] e k l :=    1 k = 1 , l = 0 , ( − 1) k +1 q 1 − k 1 − λq − 2 1 − λq − 2 k k > 1 , l = 0 , 0 otherwise , Z [ x i 0 ] e k l :=  1 k = i, l = 0 , 0 otherwise , i > 1 . Note th at R [ x 0 ] is defin ed in suc h a w a y that th e case λ = q 2 is includ ed. Note also that R [1] is in fact the charac ter ε d etermined by ε ( x n ) = 0. Any automorphism of A leav es k er ε inv arian t, so th is is a t wisted trace with THE HO CHSCHILD COHOMOLOGY RING OF T H E ST ANDARD PODLE ´ S QUAN T U M SPH E R E 7 resp ect to all automorphisms of A . Since w e h a ve for all elemen ts [ ω ] , [ η ] of the basis of H 0 ( A, σ A ) from Lemm a 8.1 Z [ ω ] [ η ] =  1 [ ω ] = [ η ] , 0 otherwise , w e can (and will) us e the R [ ω ] to determine the homology class of a giv en 0- cycle. F or example, it h elps describing the cap pr o duct actio n of the t wisted cen tre on H 0 ( A, σ A ): lemma 8.2 . The action of Λ 0 G ( A ) on H 0 ( A, σ λ A ) , i s determine d by [ x j ± 1 ] a x 0 = 0 , j > 0 , [ x i 0 ] a x 0 = [ x i +1 0 ] =  0 i = 0 , λ = q 2 k , k > 0 , − q 1 − λ q 2 − λ [ x 0 ] i = 1 , λ 6 = q 2 . Pro of. As we remark ed in (5), x 0 acts on cycles b y m ultiplication fr om the left. The claim follo ws by applying all the ab o v e constru cted twisted traces to the resu lting cycles. ✷ In particular, the s p an of the classes [ x i 0 ] ∈ H 0 ( A, σ i mod A ) is the orbit of [1] ∈ H 0 ( A, A ) under th e action of the t w isted cen tre. In a sense, the sp an of the [ x j ± 1 ] can b e viewe d as the orbit of [1] un der the cap pro d uct with x ± 1 , although the latter do not b elong to the twiste d cen tre of A : for any subalgebra B ⊂ A with σ ( B ) ⊂ B , there is a m ap H n ( B , σ B ) → H n ( A, σ A ) giv en on th e level of cycles b y the em b eddin g of B in to A in eac h tensor comp onent . If a class is in the image of this map, then taking the cap pro du ct with a twiste d cen tral elemen t of B is w ell-defined, and this app lies here to the case B is the su balgebra generated by x 1 or x − 1 , resp ectiv ely . 9. Three twisted deriv a tions The Podle ´ s sphere is a mo dule algebra ov er the Hopf dual C q [ S U (2)] ◦ of the quantised co ordinate ring of S U (2), hence t w isted pr imitiv e element s therein act as t w isted deriv ations on A . W e will not n eed Hopf algebra theory later, so we r ather state the follo w in g lemma that the reader can v erify directly b y c hec kin g compatibilit y w ith the defin in g relations of A : lemma 9.1 . The assignments ∂ 1 : x − 1 , x 0 , x 1 7→ 0 , q x − 1 , 1 + ( q + q − 1 ) x 0 , ∂ 0 : x − 1 , x 0 , x 1 7→ − x − 1 , 0 , x 1 , ∂ − 1 : x − 1 , x 0 , x 1 7→ 1 + ( q + q − 1 ) x 0 , q − 1 x 1 , 0 c an b e extende d u ni q uely to 1-c o c ycles ∂ i ∈ C 1 σ −| i | mod , ∂ i ( ab ) = σ −| i | mod ( a ) ∂ i ( b ) + ∂ i ( a ) b, a, b ∈ A, i = − 1 , 0 , 1 . The follo wing lemma describ es the cup pro d uct action of Λ 0 G ( A ) on these deriv ations. Admittedly , the result is sligh tly weird: lemma 9.2 . The Λ 0 G ( A ) -mo dule gener ate d by [ ∂ 0 ] is fr e e, b ut [ ∂ ± 1 ] ` x 0 = 0 . 8 ULRICH KR ¨ AHMER Pro of. One c hec ks directly that one h as for all a ∈ A ( ∂ ± 1 ` x 0 )( a ) = x 0 ∂ ± 1 ( a ) = ± 1 q − q − 1 ( x ∓ 1 a − ax ∓ 1 ) , so the deriv ations ∂ ± 1 ` x 0 are inn er . On the other h and, the computation of e j k x 1 − σ i mod ( x 1 ) e j k = b 1 j k in the pro of of Lemma 8.1 sho ws that no inner deriv ations in C 1 σ i mod can map x 1 to x i 0 x 1 = ( ∂ 0 ` x i 0 )( x 1 ). ✷ F or th e reason explained at the end of the previous section, it do es mak e sense to tak e the cup pro duct b et wee n ∂ ± 1 and x j ∓ 1 , although these are not t wisted central , an d this pro duces new t w isted deriv ations. Acting with them on the fundamental class and comparing the result with the generators of H 1 ( A, σ A ) compu ted in [6 ] allo ws one to describ e all t wisted deriv ations of A , see [8 ] wh ere w e carried this out f or C q [ S U (2)]. How ev er, there is little gain in this for the main pu rp ose of the present pap er w hic h is to obtain a functional describing d A in a dual fashion, so I lea ve out these calculations. Let us compute in s tead the algebra generated by the [ ∂ i ]. C lassically , a differen tial form can b e con tracted with a v ector field to redu ce its degree, and in the quantum case this is generalised by the cap p ro du ct action of (the cohomology class of ) a t wisted deriv ation on a homology class. Here is the fu ll orbit of d A und er this action of the ∂ i : d A a ∂ 0 = 2 q − 2 x − 1 x 0 ⊗ x 1 + 2 q 2 x 1 x 0 ⊗ x − 1 − 2( q 2 + q − 2 ) x 2 0 ⊗ x 0 + q − 1 x − 1 ⊗ x 1 + q x 1 ⊗ x − 1 − 2( q + q − 1 ) x 0 ⊗ x 0 , ( d A a ∂ 0 ) a ∂ 0 = 2( q 2 − q − 2 ) x 3 0 + 3( q − q − 1 ) x 2 0 , ( d A a ∂ 0 ) a ∂ − 1 = 2( − q 5 + q − 1 ) x 1 x 2 0 + ( − 2 q 2 + 1 + q − 2 1) x 1 x 0 + q − 1 x 1 , ( d A a ∂ 0 ) a ∂ 1 = 2( q − q − 5 ) x − 1 x 2 0 + ( q 2 + 1 − 2 q − 2 ) x − 1 x 0 + q x − 1 , d A a ∂ − 1 = − 2 q − 1 x 2 1 ⊗ x − 1 − 2 q − 1 x 2 0 ⊗ x 1 + 2( q 3 + q − 3 ) x 1 x 0 ⊗ x 0 − (1 + q − 2 ) x 0 ⊗ x 1 + (1 + q − 2 ) x 1 ⊗ x 0 − q − 1 ⊗ x 1 , ( d A a ∂ − 1 ) a ∂ 0 = 2( q − 3 − q 3 ) x 1 x 2 0 + ( − q 2 − 1 + 2 q − 2 ) x 1 x 0 − q − 1 x 1 , ( d A a ∂ − 1 ) a ∂ − 1 = 2( q 2 − q − 6 ) x 2 1 x 0 + ( q − 3 − q − 5 ) x 2 1 , ( d A a ∂ − 1 ) a ∂ 1 = 2( q − 8 − 1) x 3 0 + ( − q − 2 q − 1 + q − 5 + 2 q − 7 ) x 2 0 +( − 2 − q − 2 + q − 4 ) x 0 − q − 1 , d A a ∂ 1 = 2 q x 2 − 1 ⊗ x 1 + 2 q x 2 0 ⊗ x − 1 − 2( q 3 + q − 3 ) x − 1 x 0 ⊗ x 0 +( q 2 + 1) x 0 ⊗ x − 1 + ( − q 2 − 1) x − 1 ⊗ x 0 + q ⊗ x − 1 , ( d A a ∂ 1 ) a ∂ 0 = 2( q 3 − q − 3 ) x − 1 x 2 0 + (2 q 2 − 1 − q − 2 ) x − 1 x 0 − q x − 1 , ( d A a ∂ 1 ) a ∂ − 1 = 2(1 − q 8 ) x 3 0 + ( − 2 q 7 − q 5 + 2 q + q − 1 ) x 2 0 +( − q 4 + q 2 + 2) x 0 + q , ( d A a ∂ 1 ) a ∂ 1 = 2( q 6 − q − 2 ) x 2 − 1 x 0 + ( q 5 − q 3 ) x 2 − 1 . THE HO CHSCHILD COHOMOLOGY RING OF T H E ST ANDARD PODLE ´ S QUAN T U M SPH E R E 9 In homology , this reduces to: ([ d A ] a [ ∂ i ]) a [ ∂ i ] = 0 , ([ d A ] a [ ∂ 0 ]) a [ ∂ − 1 ] = − ([ d A ] a [ ∂ − 1 ]) a [ ∂ 0 ] = q − 1 [ x 1 ] , ([ d A ] a [ ∂ 0 ]) a [ ∂ 1 ] = − ([ d A ] a [ ∂ 1 ]) a [ ∂ 0 ] = q [ x − 1 ] , ([ d A ] a [ ∂ 1 ]) a [ ∂ − 1 ] = − q 2 ([ d A ] a [ ∂ − 1 ]) a [ ∂ 1 ] = ( q 2 + 1)[ x 0 ] + q . F rom this, w e see: lemma 9.3 . The ∂ i satisfy no other r elations than those dictate d by L emma 3.1, [ ∂ i ] ` [ ∂ j ] = − q 2 ij [ ∂ j ] ` [ ∂ i ] , i ≤ j. Pro of. P oincar ´ e dualit y tells that the Λ G ( A )-mod ule Ω G ( A ) is free and generated by d A , so the [ ∂ i ] satisfy u nder ` all the r elations they satisfy as linear maps on Ω G ( A ). The r esu lt follo ws f rom th e ab ov e compu tations. ✷ 10. The volume form Finally , we no w put ϕ : Ω G ( A ) → k , [ ω ] 7→ q − 1 Z [1] [ ω ] a ([ ∂ 1 ] ` [ ∂ − 1 ]) , or explicitly on c hain lev el ϕ ( a 0 , a 1 , a 2 ) = q − 1 Z [1] σ − 2 mod ( a 0 ) σ − 1 mod ( ∂ 1 ( a 1 )) ∂ − 1 ( a 2 ) . Our ab o ve computation of ([ d A ] a [ ∂ 1 ]) a [ ∂ − 1 ] implies ϕ ( d A ) = 1 , so th e functional ϕ pro vides a dual description of the f u ndamenta l class. It is a v ery useful to ol (and p robably the only really app licable one) for c hec kin g whether or not a giv en σ mod -t w isted 2-cycle has trivial class in H 2 ( A, σ mod A ) or not, wh ich as a result of th e ab ov e computations (recall that H 2 ( A, σ mod A ) ≃ C ) is the fact if and only if ϕ v anishes on the cycle. Note that w e remarke d in Secti on 8 that R [1] is actually a c haracter that we also d enote b y ε (when em b edd ing A as usual in to the quantised co ordin ate ring of S U (2), this charac ter b ecomes the restriction of the counit). Hence ϕ can also b e written as simp le as ϕ ( a 0 , a 1 , a 2 ) = q − 1 ε ( a 0 ) F ( a 1 ) E ( a 2 ) , where E , F : A → k are th e (u nt w isted) d eriv ations giv en b y E ( a ) := ε ( ∂ − 1 ( a )) , F ( a ) := ε ( ∂ 1 ( a )) . A moment’s r eflection n o w give s th e explicit formula (7) ϕ ( e ij , e k l , e mn ) = q − 1 δ i 0 δ j 0 δ k 0 δ l 1 δ m 0 δ n − 1 . This looks surpr isingly simple (not to sa y banal), usually one exp ects R to b e some sort of in tegral. Ho we v er, we observed already in [8] for the case of qu an tum S U (2) that the fun ctional app earing wh en expressing the v olume f orm du al to d A as ab ov e is giv en by something lik e the integral 10 ULRICH KR ¨ AHMER of the restriction of fun ctions to a maximal toru s whic h is a P oisson sub- group of the P oisson group qu an tised b y C q [ S U (2)]. Here w e hav e the same phenomenon, but it app ears muc h sharp er since the fun ctional R [1] is really in tegration o v er (meaning ev aluation in) a sin gle p oint . It someho w seems that the homological inform ation ab out a quantum space can b e su pp orted in a classical sub space of smaller dimension. Note also that one can alternativ ely define ϕ ± : ω 7→ ± q ∓ 1 Z [ x ∓ 1 ] ω a ( ∂ 0 ` ∂ ± 1 ) , where R [ ± 1] is the trace (no t wist) dual to [ x ± 1 ] ∈ H 0 ( A, A ). Again, the ab o v e computations giv e ϕ ± ( d A ) = 1, hence ϕ = ϕ + = ϕ − as functionals on H 2 ( A, σ mod A ) ≃ C (but not as f unctionals on C σ mod 2 ), and for some purp oses this r epresen tation of the functional migh t b e b etter su ited that ϕ . 11. Cyclic homol ogy In the classical case A = k [ X ], X a sm o oth v ariety , exterior deriv ation d turns the algebraic differentia l forms Ω( X ) into a co chain complex that computes the algebraic de Rh am cohomology H n ( X ) of X . In the non- comm utativ e case, Connes’ cyclic h omology p r o vides a s ubtle analogue of H n ( X ). The extension of cyclic homology to the t w isted co efficient s σ A arose in the w ork of Ku stermans, Murphy and T u set on co v ariant differen- tial calculi o v er q u an tum groups [13], b ut can also b e view ed as a sp ecial case of Conn es-Mosco vici’s Hopf-cyclic homology [3], see e.g. our article [8] and the references therein for more bac kground. 1 The pr ecise relation b et ween H C n ( k [ X ]) ( σ = id) and H n ( X ) is (8) H C n ( k [ X ]) ≃ Ω n ( X ) / im d ⊕ H n − 2 ( X ) ⊕ H n − 4 ( X ) ⊕ . . . , so all differen tial n -forms (not only the closed ones in ke r d !) h a v e classes in H C n ( k [ X ]), and th is giv es a map I : Ω n ( X ) ≃ H n ( k [ X ] , k [ X ]) → H C n ( k [ X ]) . F urtherm ore, d (applied to Ω n ( X ) / im d ) giv es a w ell-defined m ap B : H C n ( k [ X ]) → Ω n +1 ( X ) ≃ H n +1 ( k [ X ] , k [ X ]) whic h kills all the H n − 2 i ( X ) sum mands in (8), and finally there is S : H C n ( k [ X ]) → H C n − 2 ( k [ X ]) that cuts off the first term Ω n ( X ) / im d and lea v es the rest unto uc hed (u s ing the obvious embedd ing H n − 2 ( X ) → Ω n − 2 ( X ) / im d in th e next summand). This whole picture carries o ver to the general noncomm utativ e case an d b ecomes condensed into Connes’ S BI -sequence, see [14, 20] for the details. The upshot of this is that H C n ( A ) conta ins a wh ole lot of ballast, the really inte resting part is only the image of the n atural map I coming from Ho c hsc hild homology , wh ic h equals the kernel of the p er io dicit y map S . One wa y to defin e the cyclic theory is in terms of the op erator t : C σ n → C σ n : a 0 ⊗ . . . ⊗ a n 7→ ( − 1) n σ ( a n ) ⊗ a 0 ⊗ . . . ⊗ a n − 1 . 1 As Jac k S hapiro informed me, he will discuss in a forthcoming article th e corresp onding versi on of noncommutative de Rham theory [18]. THE HO CHSCHILD COHOMOLOGY RING OF T H E ST ANDARD PODLE ´ S QUAN T U M SPH E R E 11 Its coin v arian ts C σ n / im (id − t ) f orm a quotien t complex of ( C σ n , b ) whose homology is H C σ n ( A ) ( k sh ould conta in Q and σ should b e diagonalisable, otherwise the result might b e not what one wan ts). As a consequence, a linear f unctional ψ : C σ n → k with ψ (im b ) = 0 descends to a fun ctional on H C σ n ( A ) if it is in v arian t under t , ψ = ψ ◦ t . Clearly , ψ also induces a fun ctional on H n ( A, σ A ), but this might v anish ev en when the one on H C σ n ( A ) do esn ’t (namely when there exists χ : C σ n − 1 → k with ψ = χ ◦ b , but no suc h χ which is cyclic, χ = χ ◦ t ). Su c h a fun ctional on H C σ n ( A ) that v anishes on H n ( A, σ A ) corresp on d s to the ab o v e describ ed ballast in cyclic homology , it v anishes in the classical case on the leading term Ω n ( X ) / im d of H C n ( k [ X ]) and is rather a functional on some H C σ n − 2 k ( A ), k > 0, that is p r omoted to a fu n ctional on H C σ n ( A ) u sing th e p erio dicit y op eration S . 12. The case of the Podls ´ s sphere Let no w A b e again the Podle ´ s sphere. Sc hm ¨ udgen an d W agner ha v e constructed in [17] a non trivial cyclic 2-cocycle on A whic h later w as sho wn b y Hadfield to b e trivial when viewed on Ho c hsc hild homology . It is now natural to ask whether our volume form ϕ constructed ab o v e d o es also giv e rise to a nontrivial fu n ctional on cyclic homology . It is easily c hec k ed that a fu nctional ψ : C σ n → k v anishin g on im b is cyclic if and only if ϕ (1 , a 1 , . . . , a n ) = 0 f or all a 1 , . . . , a n ∈ A . Using this one sees that our ϕ itself is not cyclic since by (7) we ha v e ϕ (1 , x 1 , x − 1 ) = q − 1 6 = 0 . Ho wev er, w e can alter ϕ by a cob oun dary to m ak e it cyclic, so the p roblem is only a matter of represent ing the fun ctional on H 2 ( A, σ mod A ) prop erly: lemma 12.1 . D efine η := φ ◦ b : C σ mod 2 → k , wher e the line ar functional φ : A ⊗ A → k vanishes on al l e k l ⊗ e mn exc ept on the fol lowing ones: φ (1 , x 0 ) := 1 q − 2 − 1 , φ (1 , x 2 0 ) := 1 q − q − 1 , φ ( x 0 , x 0 ) := 1 2( q − q − 1 ) . Then η i nduc es the trivial functional on H 2 ( A, σ mod A ) and ϕ + η is cyclic . Pro of. By its definition, η v anish es on im b and defines the trivial func- tional on H 2 ( A, σ mod A ). By (7), the only c hain of the form 1 ⊗ e k l ⊗ e mn on whic h ϕ do es not v anish is 1 ⊗ x 1 ⊗ x − 1 , where ϕ has the v alue q − 1 , and one easily chec ks u sing η (1 , a 1 , a 2 ) = φ ( a 1 , a 2 ) − φ (1 , a 1 a 2 ) + φ ( σ mod ( a 2 ) , a 1 ) that similarly η (1 , e k l , e mn ) v anish es except when e k l ⊗ e mn = x 1 ⊗ x − 1 and then it equals − q − 1 . T he r esult follo ws. ✷ Referen ces [1] K.A. Brown, J.J. Zhang, Dualising c omplexes and twiste d Ho chschild (c o)homolo gy for no etherian Hopf algebr as , arXiv:math.RA/060373 2 (2006). [2] H. Cartan, S . Eilen b erg, Homol o gic al algebr a , Princeton Un ivers ity Press (1956). [3] A. Connes, H. Moscovici, Hopf algebr as, cyclic c ohomolo gy and the tr ansverse index the or em , Comm. Math. Phys. 198 no. 1, 199-246 (1998). 12 ULRICH KR ¨ AHMER [4] M. Dij khuizen, T. Koornwinder, Quantum homo gene ous sp ac es, duality and quantum 2 -spher es , Geom. Dedicata 52 no. 3, 291-315 ( 1994). [5] M. F arinati, Ho chschild duality, lo c alization, and smash pr o ducts , J. Algebra 284 no. 1 415-434 (2005). [6] T. Hadfield, Twiste d cycli c homolo gy of al l Po d l e ´ s quantum spher es , J. Geom. Ph ys. 57 no. 2, 339-351 (2007). [7] T. Hadfield, U. K r¨ ahmer, On the Ho chschild homolo gy of quantum SL(N) , Comptes Rendus Acad. Sci. Paris, Ser. I 343 , 9-13 (2006) [8] T. Had fi eld, U. K r¨ ahmer, Twiste d H om olo gy of Quantum SL(2) - Part II , arXiv:0711.41 02 (2007). [9] G. Ho chsc hild, B. Kostan t, A. Rosenberg, Differ ential forms on r e gular affine algebr as , T rans. AMS 102 , 383-408 (1962). [10] A. Klimyk, K. Schm ¨ udgen, Quantum gr oups and their r epr esentations , Springer (1997). [11] U. K r¨ ahmer, Poi nc ar e duali ty i n Ho chschild (c o)homolo gy , Pro ceedings of ”New tech- niques in Hopf algebras and graded ring theory” (2006). [12] U. Kr¨ ahmer, On the Ho chschild (c o)homolo gy of quantum h omo gene ous sp ac es , arXiv:0806.02 67 (2008). [13] J. 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W eib el, An intr o duction to homolo gic al algebr a , Cambridge Univ. Press (1995). University Gardens, Glasgo w G12 8QW, UK E-mail addr ess : ukraehmer@ maths.gla.ac .uk