math.RA 2015-12-18 0

Calculus structure on the Lie conformal algebra complex and the variational complex

We construct a calculus structure on the Lie conformal algebra cochain complex. By restricting to degree one chains, we recover the structure of a g-complex introduced in [DSK]. A special case of this construction is the variational calculus, for whi…

Authors: Alberto De Sole, Pedram Hekmati, Victor Kac

Calculus structure on the Lie conformal algebra complex and the v ariational complex Alb erto De Sole 1 , Pedra m Hekmati 2 , Victor G. Kac 3 1 Dipartiment o di Matematica, Univ ersit` a di Roma “La Sapienz a”, 00185 Roma, Italy E-mail: desole@mat.uniroma1.it 2 Sc hool of M athe matical Sciences, Universit y of Adelaide, Adelaide, SA 5005, Australia E-mail: pedram.hekmati@adelaide.edu.au 3 Departmen t of Mathematics, MIT, 77 Massac h usetts Av en ue, Cambridge, MA 0 2139, USA E-mail: k ac@math.mit.edu The date of receipt and acce ptance wil l be inser ted by the editor Abstract: W e construct a calculus structure on the Lie conformal a lgebra co c hain complex. By restricting to degr ee one c ha ins, w e recover the structure of a g -complex introduced in [DSK]. A sp ecial ca se of this construction is the v ariatio na l calculus , for which we provide explicit formulas. 1. Introduction A Lie conformal algebra ov er a field F , is an F [ ∂ ]-mo dule R endow ed with a bilinear map [ · λ · ] with v alues in R [ λ ], called the λ -br acket , satisfying certain sesquilinearity , skew c o mm utativ ity and Jacobi identit y . In pra ctice, λ -brack ets arise as g enerating functions for the singula r part of the op erator pro duct expan- sion in confor mal field theory [K]. Mor e r ecen tly , their domain o f a pplica bilit y has been further extended to enco de lo cal Poisson brack e ts in the theory of int egrable evolution equa tions [BDSK]. Lie confor mal algebr as resemble Lie algebra s in many wa ys and in pa r ticular their c o homology theory with co efficien ts in an R -mo dule M was dev elop ed in [BKV], [BDA K]. In [DSK], it was further shown that when the R -mo dule M is endow ed with a commutativ e asso ciative pro duct, on whic h ∂ and R act a s deriv ations, the Lie conformal algebra co chain complex ( C • ( R, M ) , d ) car ries a structure of a g -complex, where g is the L ie alg ebra of Lie conformal a lgebra 1- c ha ins. Namely , for each X ∈ g there exists a contraction op erator ι X and a Lie deriv ative L X on C • ( R, M ) satisfying the usual r ules of Cartan calculus . Moreov er, it was shown in [DSK] that in the sp ecial case of the Lie confor ma l algebra R = L i ∈ I F [ ∂ ] u i with zer o λ -brack et, acting o n an algebr a of differen tial functions V by u iλ f = X k ∈ Z + λ k ∂ f ∂ u ( k ) i , f ∈ V , (1.1) 2 A. De Sole, P . Hekmati, V. Kac the co chain complex ( C • ( R, V ) , d ) is identified with the v a riational complex, int ro duced in [GD], the Lie algebra of 1-chains for the R -mo dule V is identified with the Lie algebra of evolutionary vector fields, and the Car ta n ca lculus turns int o the v ar iational calculus. Our aim in this pap er is to extend the structure of a g -complex on C • ( R, M ) to the much richer structure o f a calculus structure. The notion o f a calculus structure origina ted in Ho chsc hild coho mology theory [DTT] (in fa c t, the defini- tion in [DTT] differs from our s by so me s igns). It is defined as a representation ( ι · , L · ) of a Gerstenhab er (= odd Poisson) alg ebra G on a complex ( Ω , d ), such that the usual Cartan’s formula ho lds L X = [ ι X , d ] . (1.2) Here ι · (resp ectiv ely L · ) is a repres en ta tion of G (resp. of G with reversed pa r it y ) viewed a s an as socia tiv e (resp. Lie) sup eralgebra . The motiv ating example of a ca lculus structure comes from differential ge- ometry . Namely , let M be a s mo oth manifold. The space of p olyvector fields Ω • ( M ), is a Gerstenhab er a lgebra, with the asso ciative pro duct giv en by the exterior multiplication ∧ , and the brack et g iven by the Schouten brack et. Then the repr e sen tation o f Ω • ( M ) on the space Ω • ( M ) of differential for ms, tog ether with the de Rham differ en tia l d , is given by the contraction o perato r ( ι X ω )( Y ) = ( − 1) p ( X )( p ( X ) − 1) 2 ω ( X ∧ Y ) , X , Y ∈ Ω • ( M ) , ω ∈ Ω • ( M ) (1.3) and the Lie deriv ative L X is given by Car tan’s for m ula (1.2). In Sectio n 2, a part from the bas ic definitions, we in tro duce the no tion of a rigged repre s en ta tion of a Lie a lg ebroid ( g , A ), whic h allows one to e x tend a structure o f a ( g , A )-complex to a calculus str ucture (Theorem 2.18 and 2.2 5). In Section 3, for a n y Lie alg ebra g a nd a g -mo dule A , where A is a commuta- tive ass ociative alg e br a on which g acts b y deriv ations, w e c o nstruct a calculus structure ( ∆ • ( g , A ) , ∆ • ( g , A )), where ∆ • ( g , A ) is the spa c e o f Lie algebra chains endow ed with a s tructure of a Gerstenhab er alge br a, and ( ∆ • ( g , A ) , d ) is the complex of Lie alg ebra co c hain (Theorem 3.1). Keeping in mind the annihilation Lie algebr a of a Lie conformal alg ebra, we construct a “topo lo gical” calculus structure in the case when g is a linearly co mpact Lie algebra . In Section 4 we introduce a Gerstenhab er algebr a structure on the spa ce of Lie conformal a lgebra chains C • ( R, M ) for an arbitr ary mo dule M with a commutativ e as s ociative algebra structure over a Lie conformal algebra R , act- ing on M by deriv a tions. This extends the Lie algebr a structure on the space of 1-chains with reversed parity , g = Π C 1 ( R, M ), defined in [DSK, The o rem 4.8]. This allows us to extend the g -structure on the complex of Lie conformal algebra co chains C • ( R, M ) c o nstructed in [DSK] to a calculus struc tur e (Theo- rem 4.11). F urthermor e, we co nstruct a morphis m from the topo logical calculus structure for the (linearly co mpact) annihilation Lie a lgebra Lie − R o f a finite Lie confor ma l algebra R to the calculus structure ( C • ( R, M ) , C • ( R, M )) which induces a n isomo rphism of the reduced b y ∂ former calculus structure to the torsionless pa rt of the la tter calculus structure (Theorem 4.14 ), extending that in [DSK] for g -structures. This is us e d in Section 5 to identify the v a r iational complex Ω • ( V ) over an alg ebra of differential functions V on ℓ differential v a ri- ables, with the complex C • ( R, V ), where R is the free F [ ∂ ]-module of rank ℓ Calculus structure 3 with zero λ -brack et, acting on V via (1.1), and to extend the identification of g -structures obtained in [DSK], to an e x plicit constr uction of the v aria tio nal calculus str uc tur e ( Ω • ( V ) , Ω • ( V )), where Ω • ( V ) is the Gerstenhab er algebra of all evolutionary p olyvector fields ov e r V . Throughout the pap er all vector spaces a re considere d ov er a field F of char- acteristics zer o. Unless otherwise s pecified, direct sums and tenso r pro ducts are considered ov er F . 2. Calcul us structure on a complex In this s ection we introduce the basic definitions of a Gers tenhaber a lgebra a nd of a ca lculus structur e, a nd pr o ve so me simple related res ults that will b e used throughout the pa p er. 2.1. Rigge d r epr esentations of Lie sup er algebr as. Re c a ll that a vector supe rspace is a Z / 2 Z -gra ded vector spa ce V = V ¯ 0 ⊕ V ¯ 1 . If a ∈ V α , where α ∈ Z / 2 Z = { ¯ 0 , ¯ 1 } , one says that a has parity p ( a ) = α . One denotes by Π V the super space obtained from V b y reversing the parity , na mely Π V = V as a vector s pace, with pa rit y ¯ p ( a ) = p ( a ) + ¯ 1. An endomo r phism of V is called even (res p. o dd) if it pres erv es (resp. reverses) the parity . The s uperspa ce E nd ( V ) of a ll endo morphisms of V is endow ed with a Lie sup eralgebra structure by the formula: [ A, B ] = A ◦ B − ( − 1) p ( A ) p ( B ) B ◦ A . Definition 2.1 . A r epr esentation of a Lie sup er algebr a g on a ve ctor su p ersp ac e V , X 7→ L X ∈ End( V ) , is c al le d rigged if it is endowe d with an even line ar map ι · : Π g → End( V ) ( i.e. a p arity r eversing map g → E nd( V ) ), denote d X 7→ ι X , such that: (i) [ ι X , ι Y ] = 0 for al l X , Y ∈ g , (ii) [ L X , ι Y ] = ι [ X,Y ] for al l X , Y ∈ g . Throughout the pap er we will denote the par it y of the Lie sup eralgebra g b y ¯ p . Hence, for the line a r map ι · : Π g → End( V ), we hav e p ( ι X ) = ¯ p ( X ) + ¯ 1. Recall that a c omplex ( Ω , d ) is a vector sup erspace Ω , endow ed with a n o dd endomo rphism d ∈ End( Ω ) s uc h that d 2 = 0 . A representation o f a Lie sup e ralgebra g on a complex ( Ω , d ) is a representation of g on the sup erspace Ω , denoted X 7→ L X ∈ End( Ω ), such that [ L X , d ] = 0. Recall a lso (see e.g. [DSK]) tha t a g - c omplex is a pair ( g , Ω ), wher e g is a Lie sup e ralgebra, ( Ω , d ) is a co mplex, endow ed with a linear map ι · : Π g → End( Ω ), satisfying the following conditions: ( i ) [ ι X , ι Y ] = 0 for all X , Y ∈ g , ( ii ) [[ ι X , d ] , ι Y ] = ι [ X,Y ] for a ll X ∈ g . This is also called a g - structur e on the complex ( Ω , d ). Lemma 2.2 . Any g -c omplex ( g , Ω ) gives rise to a rigge d r epr esentation of the Lie sup er algebr a g on t he c omplex ( Ω , d ) , obtaine d by defining t he map L · : g → End( Ω ) by Cartan’s for m ula : L X = [ ι X , d ] . (2.1) 4 A. De Sole, P . Hekmati, V. Kac Pr o of. Indeed, conditions ( i ) and ( ii ) in Definition 2.1 co incide , via Cartan’s formula, with co nditio ns ( i ) and ( ii ) ab o ve. Mor eo ver, since d 2 = 1 2 [ d, d ] = 0, we immediately get that [ L X , d ] = 0 for every X ∈ g . Finally we hav e , by Ca rtan’s formula (2.1) and condition ( ii ), [ L X , L Y ] = [ L X , [ ι Y , d ]] = [[ L X , ι Y ] , d ] ± [ ι Y , [ L X , d ]] = [ ι [ X,Y ] , d ] = L [ X,Y ] . ⊓ ⊔ 2.2. Rigge d r epr esentations of Lie algebr oids. Recall that a Lie sup er algebr oid is a pair ( g , A ), wher e g is a Lie super algebra, A is a commutativ e asso ciative algebra, such that g is a left A -mo dule and A is a left g -mo dule, sa tisfying the following compatibility co nditions ( X , Y ∈ g , f , g ∈ A ): ( i ) ( f X )( g ) = f ( X ( g )), ( ii ) X ( f g ) = X ( f ) g + f X ( g ), ( iii ) [ X , f Y ] = X ( f ) Y + f [ X , Y ]. R emark 2.3. Since we assume A to b e pure ly even, the o dd pa rt of g necessa r ily acts trivia lly on A . One can consider also A to be a co mm utative asso ciative sup e ralgebra, but then the s ig ns in the for m ula s b ecome mor e complicated. Example 2.4. If A is a c o mm uta tive asso ciative algebra and g is a subalg e br a of the L ie algebra of deriv ations o f A such that A g ⊂ g , then, o b viously , ( g , A ) is a Lie algebroid. Example 2.5. If g is a Lie super algebra with parity ¯ p , a cting by deriv a tions o n a commutativ e asso ciative sup eralgebr a A , then ( A ⊗ g , A ) is a Lie alge broid with Lie brack et [ f ⊗ X , g ⊗ Y ] = f g ⊗ [ X , Y ] + f X ( g ) ⊗ Y − ( − 1) ¯ p ( X ) ¯ p ( Y ) g Y ( f ) ⊗ X . Example 2.6. Given a Lie sup eralgebr oid ( g , A ), we can co nstruct tw o Lie sup er- algebras : g ⋉ A and g ⋉ Π A , with Lie bra c k et which extends that on g by letting, for f , g ∈ A and X ∈ g , [ f , g ] = 0, [ X , f ] = X ( f ) and [ f , X ] g iv en by skew co m- m utativity . Both these Lie super algebras give ris e to Lie super algebroids in the obvious way . Definition 2.7 . (a) A representation of a Lie sup er algebr oid ( g , A ) on a ve ctor sup ersp ac e V is a left A -mo dule st ructur e on V , denote d f 7→ ι f , t o gether with a left g - m o dule struct u r e on V , denote d X 7→ L X , such t hat, for X ∈ g , f ∈ A , we have [ L X , ι f ] = ι X ( f ) . (b) Given ǫ ∈ F , an ǫ - r igged repr e sen tation of a Lie sup er algebr oid ( g , A ) (with p arity of g denote d by ¯ p ) on a ve ctor s u p ersp ac e V is a left A -mo dule structu re on V , ι · : A → End( V ) , to gether with a rigge d r epr esent ation of the Lie sup er algebr a g ⋉ Π A , define d by t he line ar maps ι · : Π g ⊕ A → End( V ) , L · : g ⊕ Π A → End( V ) , satisfying the fol lowing c omp atibility c onditions ( f , g ∈ A, X ∈ g ): (i) L f g = L f ι g + ι f L g , (ii) ι f X = ι f ι X , (iii) L f X = ι f L X − ( − 1) ¯ p ( X ) L f ι X − ǫι X ( f ) . Calculus structure 5 R emark 2.8. If ( g , A ) is a Lie sup eralgebro id, then b oth A and g are ( g , A )- mo dules (but in gener al they are not rigged). O n the other hand, a s we will s ee in Pro position 2.14, they extend to a 1-rigg ed r epresent ation o f the Lie s uperal- gebroid ( g , A ) on the vector sup erspace S A ( Π g ). Ho wev er , for the a pplications to calculus structure , the mo st impo rtan t role will b e played by the 0-rig ged re pr e- sentations. Indeed, as we will s ee in Pr opositio n 2.11 b elow, a n y ( g , A )-complex gives ris e to a 0-r igged repr esen tation of the Lie sup eralgebro id ( g , A ). Definition 2.9 . (a) A representation of a Lie sup er algebr oid ( g , A ) on a complex ( Ω , d ) is a r epr esentation of ( g , A ) on the ve ctor sup ersp ac e Ω such t hat [ L X , d ] = 0 for every X ∈ g . (b) A ( g , A ) - complex ( Ω , d ) , wher e ( g , A ) is a Lie sup er algebr oid, is a g ⋉ Π A - c omplex (for the Lie sup er algebr a g ⋉ Π A in Example 2.6) s u ch that the line ar map ι · : Π g ⊕ A → End( Ω ) satisfies t he fol lowing two addi tional c onditions (for f , g ∈ A, X ∈ g ): (i) ι f g = ι f ι g , (ii) ι f X = ι f ι X . The following result allows us to extend a g -complex to an ( A ⊗ g , A )-complex. Lemma 2.1 0. L et A b e a c ommut ative asso ciative algebr a and let g b e a Lie sup er algebr a, with p arity ¯ p , acting on A by derivations, so t hat we have t he c orr esp onding Lie sup er algebr oid ( A ⊗ g , A ) fr om Example 2.5. L et ( Ω , d ) b e a c omplex endowe d with a structu r e of a g -c omplex, ι · : Π g → E nd( Ω ) , and with a structur e of a left A -mo dule, denote d by ι · : A → End( Ω ) . Define the m ap L · : g ⊕ Π A → E nd( Ω ) by Cartan ’s formula: L a = [ ι a , d ] , a ∈ g ⊕ Π A . A ssume that the fol lowing c onditions hold: (i) [ ι X , ι f ] = 0 for al l f ∈ A, X ∈ g , (ii) [ L f , ι g ] = 0 , for al l f , g ∈ A , (iii) [ L X , ι f ] = ι X ( f ) , for al l f ∈ A, X ∈ g . Then, we have a structu r e of an ( A ⊗ g , A ) -c omplex on ( Ω , d ) by letting ι f ⊗ X = ι f ι X , for f ∈ A and X ∈ g . Pr o of. By definition o f a co mplex over the Lie sup eralgebroid ( A ⊗ g , A ), we need to prove that the fo llowing r elations hold: (1) [ ι a , ι b ] = 0, for a, b ∈ ( A ⊗ Π g ) ⊕ A , (2) [ L a , ι b ] = ι [ a,b ] , for a, b ∈ ( A ⊗ g ) ⋉ Π A , (3) ι f g = ι f ι g , for f , g ∈ A , (4) ι f ( g ⊗ X ) = ι f ι g ⊗ X , fo r f , g ∈ A, X ∈ Π g , where L a , a s be fo re, is defined by Ca r tan’s form ula for a ∈ ( A ⊗ g ) ⊕ Π A . Relatio n (1) is immedia te by the definition of ι f ⊗ X and ass umption ( i ). Re la tion (3 ) holds by the a ssumption that ι : A → End( Ω ) defines a structure of a left A -mo dule. Relation (4) is also immediate. W e are left to pr o ve re la tion (2). When a, b ∈ Π A , it holds by a s sumption ( ii ). When a = f ∈ Π A, b = g ⊗ X ∈ A ⊗ g , it follows by a straightforward computation using the following identit y , [ L f , ι X ] = − ( − 1) ¯ p ( X ) ι X ( f ) , 6 A. De Sole, P . Hekmati, V. Kac which can b e easily check ed. Finally , when a = f ⊗ X ∈ A ⊗ g a nd b = g ∈ Π A or b = g ⊗ Y ∈ A ⊗ g , r e lation (2) follows using the identit y , L f ⊗ X = ι f L X − ( − 1) ¯ p ( X ) L f ι X , which is ag ain str a igh tforward to check. ⊓ ⊔ The following result gener alizes Lemma 2 .2 to the case o f Lie sup eralgebroids. Prop osition 2.11. Any ( g , A ) -c omplex ( Ω , d ) gives rise t o a 0 -rigge d r epr esen- tation of the Lie sup er algebr oid ( g , A ) on the ve ctor sup ersp ac e Ω , obtaine d by defining the map L · : g ⋉ Π A → End( Ω ) by Cartan ’s formula: L a = [ ι a , d ] for al l a ∈ g ⋉ Π A . Pr o of. Co ndition ( i ) in Definition 2.9(b) gua rant ees that Ω is a left A -mo dule. By Lemma 2 .2 we know that the map L · : g ⋉ Π A → End( V ), g iv en b y Car- tan’s formula, is a Lie sup eralgebra homomor phis m. Moreov er , c o ndition ( ii ) in Definition 2.7(b) coincides with condition ( i i ) in Definition 2.9(b). Hence, to conclude the pro of, we are left to chec k tha t ι · and L · satisfy the compatibility conditions ( i ) and ( ii i ) in Definition 2.7(b). Both o f them follow immediately by Cartan’s for m ula . ⊓ ⊔ Example 2.12. Let A b e the algebra of smoo th functions on a smo oth manifold M , g be the Lie algebra of smo oth vector fields on M , and Ω b e the complex of smo oth differential forms on M with the de Rham differential d . Then ( g , A ) is a Lie a lg ebroid. Mo reov er, the map ι · : Π g ⊕ A → End( V ), where ι f is the m ultiplication b y f ∈ A , and ι X is the contraction op erator by the v ector field X ∈ g , defines a structur e of a ( g , A )-complex on Ω . Hence, by Prop osition 2 .11, we get a 0-rig ged representation of the Lie a lg ebroid ( g , A ) on the complex Ω , where L f is the m ultiplication by − d f in the alg ebra Ω , for f ∈ A , and L X is the Lie der iv ative by the vector field X ∈ g . 2.3. Gerstenhab er ( = o dd Poisson) algebr as. Recall that, given a commutativ e asso ciative algebr a A , and an A -mo dule structure on a vector sup e rspace V , the symmetric, (resp ectively exterio r) sup eralgebra S A ( V ) (resp. V A ( V )) is defined as the quotient of the tensor sup eralgebr a T A ( V ) by the relations u ⊗ A v − ( − 1) p ( u ) p ( v ) v ⊗ A u (re s p. u ⊗ A v + ( − 1) p ( u ) p ( v ) v ⊗ A u ). Note that S A ( Π V ) is the same a s V A V as an A -mo dule (but not as a vector sup erspace). Definition 2.1 3. A Gers tenhaber algebr a (also known as an odd Poisson a lg e- bra ) is a ve ctor sup ersp ac e G , with p arity p , endowe d with a pr o duct ∧ : G ⊗ G → G , and a br acket [ · , · ] : G ⊗ G → G satisfying t he fol lowing pr op erties: (i) ( G , ∧ ) is a c ommutative asso ciative sup er algebr a, (ii) ( Π G , [ · , · ]) is a Lie sup er algebr a, (iii) the fol lowing left L eibniz rule holds: [ X , Y ∧ Z ] = [ X , Y ] ∧ Z + ( − 1 ) ( p ( X )+ ¯ 1) p ( Y ) Y ∧ [ X , Z ] . (2.2) Calculus structure 7 F rom the left Leibniz rule (2.2) and skew c omm uta tivit y , we g et the right Le ibniz rule : [ X ∧ Y , Z ] = X ∧ [ Y , Z ] + ( − 1) p ( Y )( p ( Z )+ ¯ 1) [ X , Z ] ∧ Y . (2.3) Prop osition 2.14. L et ( g , A ) b e a Lie su p er algebr oid. Then ther e exists a un ique structur e of a Gerstenhab er algebr a on the sup ersp ac e G = S A ( Π g ) , with p arity denote d by p , wher e the c ommutative asso ciative sup er algebr a pr o duct ∧ on G is the pr o duct in the symmetric sup er algebr a S A ( Π g ) , and the Lie sup er algebr a br acket [ · , · ] on Π G , c al le d the Schouten brack e t , extends inductively that on t he Lie sup er algebr a g ⋉ Π A fr om Example 2.6 by t he L eibniz rule (2.2) . Pr o of. The symmetric super algebra S A ( Π g ) is defined as the quotient of the tensor sup eralgebr a T ( Π g ⊕ A ) by the tw o-sided ideal K gener ated by the rela- tions ( i ) a ⊗ b = ( − 1) p ( a ) p ( b ) b ⊗ a , a, b ∈ Π g ⊕ A , ( ii ) f ⊗ X = f X , f ∈ A, X ∈ Π g . (2.4) Therefore, in o rder to prov e that the Schouten bra c ket is well defined, w e need to do three things. First, we chec k that its inductiv e definitio n preserves asso cia- tivit y of the tensor pr oduct, so that we hav e a well-defined br a c k et o n the whole tensor a lgebra, [ · , · ] e : Π T ( Π g ⊕ A ) × Π T ( Π g ⊕ A ) → Π S A ( Π g ). Seco nd, w e argue that, in order to prov e that K is in the kernel of this brack et, it suffice s to show that it preser v es rela tions (2.4)( i ) and ( ii ). Finally , we pr o ve that thes e relations a r e indeed preserved. W e start by defining a br ac ket [ · , · ] e : Π T ( Π g ⊕ A ) × Π T ( Π g ⊕ A ) → Π S A ( Π g ), such that its restriction to g ⊕ Π A co incides with the g iv en Lie brack et on g ⋉ Π A . W e do it, inductively , in three steps . First we extend it to a brack et [ · , · ] e : ( g ⊕ Π A ) × Π T ( Π g ⊕ A ) → Π T ( Π g ⊕ A ), by the left Leibniz rule (2.2) with ∧ r eplaced by ⊗ and [ · , · ] r e pla ced by [ · , · ] e . T o prov e that this ma p is well defined we chec k that the left Leibniz rule preser v es the asso ciativity r e la - tion in the tensor algebr a. Indeed, b o th [ X , Y ⊗ ( Z ⊗ W )] e and [ X , ( Y ⊗ Z ) ⊗ W ] e are equal to [ X , Y ] e ⊗ Z ⊗ W + ( − 1 ) ( p ( X )+1) p ( Y ) Y ⊗ [ X , Z ] e ⊗ W +( − 1) ( p ( X )+1)( p ( Y )+ p ( Z )) Y ⊗ Z ⊗ [ X , W ] e . (2.5) W e then further extend it to a bracket [ · , · ] e : Π T ( Π g ⊕ A ) × Π T ( Π g ⊕ A ) → Π T ( Π g ⊕ A ), by the rig h t Leibniz rule (2.3), with the same c hanges in notatio n. Again, we prove that this map is well defined by c hecking that the right Leibniz rule preserves asso ciativity . Indeed, bo th [ X ⊗ ( Y ⊗ Z ) , W )] e a nd [( X ⊗ Y ) ⊗ Z, W ] e are equal to X ⊗ Y ⊗ [ Z , W ] e + ( − 1) p ( Z )( p ( W )+1) X ⊗ [ Y , W ] e ⊗ Z +( − 1) ( p ( Y )+ p ( Z ))( p ( W )+1) [ X , W ] e ⊗ Y ⊗ Z . (2.6) Finally , we co mpose the bra c k et [ · , · ] e with the canonical quotient map Π T ( Π g ⊕ A ) → Π S A ( Π g ), and we k eep the same no tation for the resulting map: [ · , · ] e : Π T ( Π g ⊕ A ) × Π T ( Π g ⊕ A ) → Π S A ( Π g ). W e claim that this map satisfies b oth the left and the right Leibniz rules. The right Leibniz rule holds by co nstruction, while for the left one we have to chec k that, when computing [ X ⊗ Y , Z ⊗ W ], we get the same res ult if we first apply the left Leibniz r ule a nd then the r igh t 8 A. De Sole, P . Hekmati, V. Kac one, or vice versa. As the reader ca n easily chec k , the results are not eq ua l in the tensor algebra, but they b ecome equal after w e pass to the symmetric a lgebra. Next, it is immediate to check that the bra c ket [ · , · ] e pre serves the r elations (2.4)( i ) a nd ( ii ), namely , the differences b et ween the LHS and RHS in b oth relations lie in the center of this br a c k et. This allows us to conclude, r e c alling (2.5) and (2 .6 ), that tw o -sided idea l K ⊂ T ( Π g ⊕ A ) generated by the r elations (2.4)( i )and ( ii ) is in the center of the bracket [ · , · ] e . Hence it factor s throug h a well defined bracket [ · , · ] : S A ( Π g ) × S A ( Π g ) → S A ( Π g ) s atisfying both the left and the rig h t Leibniz r ules (2 .2) and (2 .3 ). Using this it is easy to chec k , by induction, that the brack et is skew commut ative, and after that, ag ain b y induction, that it satisfies the Jacobi identit y . ⊓ ⊔ R emark 2.15. If g is a Lie sup eralgebra with pa rit y ¯ p , the corr esponding parity p in the Gerstenhab er alg ebra G = S A ( Π g ) is p ( X 1 ∧ · · · ∧ X m ) = ¯ p ( X 1 ) + · · · + ¯ p ( X m ) + m , (2.7) and the parity ¯ p of the Lie sup eralgebra Π G is ¯ p ( X 1 ∧ · · · ∧ X m ) = ¯ p ( X 1 ) + · · · + ¯ p ( X m ) + m + 1 . (2.8) One derives from the left and right Le ibniz rules (2.2) and (2.3) explicit formulas for the Schouten brack et b et ween tw o arbitrary ele ments of the Gerstenhab er algebra G . F o r f ∈ A and X = X 1 ∧ · · · ∧ X m ∈ G , with X i ∈ g , we hav e [ f , X 1 ∧ · · · ∧ X m ] = ( − 1 ) ¯ p ( X 1 )+ ··· + ¯ p ( X m )+ m [ X 1 ∧ · · · ∧ X m , f ] = m X i =1 ( − 1) ¯ p ( X 1 )+ ··· + ¯ p ( X i − 1 )+ i X i ( f ) X 1 ∧ i ˇ · · · ∧ X m , (2.9) while for X = X 1 ∧ · · · ∧ X m , Y = Y 1 ∧ · · · ∧ Y n ∈ G , with X i , Y j ∈ g , we hav e  X 1 ∧ · · · ∧ X m , Y 1 ∧ · · · ∧ Y n  = m X i =1 n X j =1 ( − 1) s ij ( X,Y ) [ X i , Y j ] ∧ X 1 ∧ i ˇ · · · ∧ X m ∧ Y 1 ∧ j ˇ · · · ∧ Y n , (2.10) where s ij ( X, Y ) =  ¯ p ( X i ) + 1  ¯ p ( X 1 ) + · · · + ¯ p ( X i − 1 ) + i + 1  +  ¯ p ( Y j ) + 1  ¯ p ( Y 1 ) + · · · + ¯ p ( Y j − 1 ) + j + 1  + ¯ p ( Y j )  ¯ p ( X 1 )+ i ˇ · · · + ¯ p ( X m ) + m + 1  . In pa rticular, if g is a Lie algebra, then s ij ( X, Y ) = i + j . Calculus structure 9 2.4. R epr esentations of a Gerstenhab er algebr a. Definition 2.1 6. A repres en ta tion of a Gerstenhab er algebr a G with p arity p on a sup ersp ac e V is a mo dule structur e over the c ommutative asso ciative s up er al- gebr a ( G , ∧ ) , denote d by ι · : G ⊗ V → V , X ⊗ v 7→ ι X ( v ) , and c al le d contraction , to gether with a mo dule structu r e over t he Lie sup er algebr a ( Π G , [ · , · ]) , denote d by L · : G ⊗ V → V , X ⊗ v 7→ L X ( v ) , and c al le d Lie deriv ative , su ch that the left L eibniz ru le is pr eserve d: [ L X , ι Y ]  = L X ι Y − ( − 1) ( p ( X )+ ¯ 1) p ( Y ) ι Y L X  = ι [ X,Y ] . (2.11) F or ex a mple, letting ι X = X ∧ and L X = a d X , we get a repre s en ta tion of a Gerstenhab er alg ebra G on itself, called its adjoint repres en ta tion. R emark 2.17. Note that a representation of a Ger stenhaber alge bra ( G , ∧ , [ · , · ]) on V is the sa me a s a rig g ed repres en ta tion of the Lie sup eralge br a ( Π G , [ · , · ]) such that the rigging X 7→ ι X is a repr esen tation of the asso ciative super algebra ( G , ∧ ). Theorem 2.18. L et ( g , A ) b e a Lie sup er algebr oid, and c onsider the Gersten- hab er algebr a G = S A ( Π g ) , with p arity p . Then any ǫ -rigge d r epr esentation of the Lie sup er algebr oid ( g , A ) on a ve ctor sup ersp ac e V , extends uniquely to a r ep- r esentation of the Gerstenhab er algebr a G on V s u ch that, for every X , Y ∈ G , the fol lowing ǫ - r igh t Leibniz rule holds: L X ∧ Y = ι X L Y + ( − 1) p ( Y ) L X ι Y − ǫ ( − 1) p ( Y ) ι [ X,Y ]  = ι X L Y + ( − 1) p ( X ) p ( Y ) ι Y L X + (1 − ǫ )( − 1) p ( Y ) ι [ X,Y ]  . (2.12) Pr o of. Since the contraction ι · : G → E nd ( V ) is a representation of the com- m utative as socia tiv e sup eralg e br a ( G , ∧ ), and it extends the rigg ing of the r ep- resentation of the Lie sup eralgebra g ⋉ Π A on V , it is for c ed to b e given by the following formula: ι X 1 ∧···∧ X m = ι X 1 · · · ι X m , (2.13) for all X 1 , . . . , X m ∈ g . It is immediate to c heck, using the assumptions that ι f X = ι f ι X for all f ∈ A, X ∈ g , a nd [ ι a , ι b ] = 0 for a ll a, b ∈ Π g ⊕ A , that the contraction map is a well-defined representation of the commutativ e a ssocia tiv e sup e ralgebra ( G , ∧ ). By assumption, the Lie deriv ative L · : Π G → End( V ) is defined by extending, inductively , the repr e s en tation of the Lie sup eralgebra g ⋉ Π A on V , using equation (2.12 ) . In or der to prov e that the map L · is well defined, we pr oceed a s in the pro of of Prop osition 2.14. First, we define a map e L · from the tensor algebra T ( Π g ⊕ A ) to E nd( V ) (which reverses the pa rit y), e x tending L · : g ⊕ Π A → End( V ), inductively , by saying that e L X ⊗ Y is given by the RHS in (2.12). By applying (2.12) twice, we g et tha t b oth e L X ⊗ ( Y ⊗ Z ) and e L ( X ⊗ Y ) ⊗ Z are equal to ι X ι Y e L Z + ( − 1) p ( Z ) ι X e L Y ι Z + ( − 1) p ( Y )+ p ( Z ) e L X ι Y ι Z − ǫ ( − 1) p ( Z )  ι X ι [ Y ,Z ] + ( − 1) p ( Y ) ι [ X,Y ] ι Z + ( − 1) p ( X ) p ( Y ) ι Y ι [ X,Z ]  , (2.14) proving that e L · preserves the as socia tivit y relation for the tensor pro duct. Ab o ve we deno ted, by an abuse of notation, the lifts of the contraction map and the 10 A. De Sole, P . Hekmati, V. Kac Schouten bracket to the tenso r a lgebra T ( Π g ⊕ A ) b y ι · and [ · , · ] res pectively . Hence e L · is a well defined map: T ( Π g ⊕ A ) → E nd ( V ). Moreover, the fact that e L · preserves the defining relations (2.4 )( i ) and ( ii ) is enco ded in the a s- sumption that V is an ǫ - r igged representation of the Lie sup eralgebroid ( g , A ). More pre cisely , for the relation (2.4)( i ) with a = f , b = g ∈ A , we hav e that e L f ⊗ g = ι f L g + L f ι g , which is the same as e L g ⊗ f thanks to co ndition ( i ) in Def- inition 2.7(b) and the fact that L f and ι g commute. F or a = X, b = Y ∈ g , we have e L X ⊗ Y − ( − 1) p ( X ) p ( Y ) e L Y ⊗ X = [ ι X , L Y ] + ( − 1) p ( Y ) [ L X , ι Y ] − ǫ  ι [ X,Y ] + ( − 1) 1+( p ( X )+ ¯ 1)( p ( Y )+ ¯ 1) ι [ Y ,X ]  , a nd this is zero b y the definition of ǫ -rigged repre- sentation and by the skewcomm utativity of the Lie br ac ket on g . Finally , when a = X ∈ g , b = f ∈ A , we hav e tha t b oth e L f ⊗ X and e L X ⊗ f are equal to ι f L X + ( − 1) p ( X ) L f ι X − ǫι X ( f ) , thanks to the a ssumption that [ L X , ι f ] = ι X ( f ) . Moreov er, this expressio n is equal to L f X , by condition ( iii ) in Definition 2.7(b), th us proving that e L · preserves the relation (2.4)( ii ). What w e just pr o v ed a l- lows us to co nclude that tw o-sided ideal K ⊂ T ( Π g ⊕ A ) g enerated by the relations (2.4)( i ) and ( ii ) is in the kernel of e L · (this is not immedia te s inc e e L · is not a homomorphism of a ssoc ia tiv e algebra s). Indeed, b oth the c on traction map ι · and the Schouten brack e t [ · , · ] are de fined on the sy mmetric sup eral- gebra S A ( Π g ), and hence, when lifted to the tensor algebr a T ( Π g ⊕ A ), they map the ideal K to zero. Therefore it is cle ar, from the expressio n (2.14) for e L X ⊗ Y ⊗ Z , tha t K is in the kernel o f e L · . Hence e L · factors thro ug h a well defined map L · : S A ( Π g ) → End( V ). T o complete the pro of we hav e to chec k that the pa ir ( ι · , L · ) is a Gersten- hab er algebra re pr esen tation. By as sumption the left Leibniz rule (2.11) holds for X , Y ∈ Π g ⊕ A . Ther efore, in order to prove (2.1 1) by induction, w e note that, [ L X , ι Y ∧ Z ] = [ L X , ι Y ι Z ] = [ L X , ι Y ] ι Z + ( − 1) ( p ( X )+ ¯ 1) p ( Y ) ι Y [ L X , ι Z ] = ι [ X,Y ] ι Z + ( − 1) ( p ( X )+ ¯ 1) p ( Y ) ι Y ι [ X,Z ] = ι [ X,Y ∧ Z ] , for a ll X , Y , Z ∈ G such that Y , Z hav e deg ree at lea st 1, and [ L X ∧ Y , ι Z ] = [ ι X L Y + ( − 1) p ( Y ) L X ι Y − ǫ ( − 1) p ( Y ) ι [ X,Y ] , ι Z ] = ι X [ L Y , ι Z ] + ( − 1) p ( Y )( p ( Z )+ ¯ 1) [ L X , ι Z ] ι Y = ι X ι [ Y ,Z ] + ( − 1) p ( Y )( p ( Z )+ ¯ 1) ι [ X,Z ] ι Y = ι [ X ∧ Y ,Z ] , for all X , Y , Z ∈ G such that X, Y hav e degree at least 1. In the computatio ns ab o ve we used the inductive as sumptions, for m ula (2.12 ) , and the commutation relation [ ι X , ι Y ] = 0. Finally , we use the above r e sults to prov e, by induction, that L · : Π G → End( V ) is a Lie sup eralge bra ho momorphism: [ L X , L Y ] = L [ X,Y ] for all X , Y ∈ Π G . If b oth X, Y a re in g ⊕ Π A , this ho lds b y as sumption. Moreov er, by skew- commutativit y , it suffices to check the homomorphism co ndition for X , Y ∧ Z , where b oth Y a nd Z have degr ee gr eater or equal than 1: [ L X , L Y ∧ Z ] = L [ X,Y ∧ Z ] . (2.15) Calculus structure 11 The LHS of (2.15) is, b y inductive as s umption, [ L X , L Y ∧ Z ] = [ L X , ι Y L Z + ( − 1) p ( Z ) L Y ι Z − ǫ ( − 1) p ( Z ) ι [ Y ,Z ] ] = [ L X , ι Y ] L Z + ( − 1) ( p ( X )+ ¯ 1) p ( Y ) ι Y [ L X , L Z ] + ( − 1) p ( Z ) [ L X , L Y ] ι Z +( − 1) p ( Z )+( p ( X )+ ¯ 1)( p ( Y )+ ¯ 1) L Y [ L X , ι Z ] − ǫ ( − 1) p ( Z ) [ L X , ι [ Y ,Z ] ] = ι [ X,Y ] L Z + ( − 1) ( p ( X )+ ¯ 1) p ( Y ) ι Y L [ X,Z ] + ( − 1) p ( Z ) L [ X,Y ] ι Z +( − 1) p ( Z )+( p ( X )+ ¯ 1)( p ( Y )+ ¯ 1) L Y ι [ X,Z ] − ǫ ( − 1) p ( Z ) ι [ X, [ Y , Z ]] . Similarly , the RHS of (2.1 5) is L [ X,Y ∧ Z ] = L [ X,Y ] ∧ Z + ( − 1) ( p ( X )+ ¯ 1) p ( Y ) L Y ∧ [ X ,Z ] = ι [ X,Y ] L Z + ( − 1) p ( Z ) L [ X,Y ] ι Z − ǫ ( − 1) p ( Z ) ι [[ X,Y ] , Z ] +( − 1) ( p ( X )+ ¯ 1) p ( Y )  ι Y L [ X,Z ] + ( − 1) p ( Z )+( p ( X )+ ¯ 1) L Y ι [ X,Z ] − ǫ ( − 1) p ( X )+ p ( Z )+1 ι [ Y , [ X, Z ]]  . Equation (2.15) now follows by the Ja cobi identit y for the Schouten bra c ket. ⊓ ⊔ R emark 2.19. One can s ho w that among all p o ssible express io ns fo r L X ∧ Y of the form aι X L Y + bι Y L X + cL X ι Y + dL Y ι X + eι [ X,Y ] , with a, b, c, d, e ∈ F , only those given by (2 .12) satisfy the left Le ibniz rule (2.1 1), and therefore g iv e rise to a repres e ntation of the Gerstenhab er algebra S A ( Π g ). Example 2.20. The adjoint repr esen tation o f the Ger stenhaber algebra S A ( Π g ) on itself, satisfies the ǫ -right Le ibniz for m ula with ǫ = 1. In the next s ubsection we will see how to construct representations of the Gerstenhab er algebra S A ( Π g ) satisfying the ǫ -right Leibniz formula with ǫ = 0 , starting from a ( g , A )-co mplex and using Ca rtan’s formula. Example 2.21. If a Lie sup eralgebroid ( g , A ) is s uch that the a c tio n of g on A is trivial, then every ǫ 0 -rigged repr esen tation of ( g , A ) on a v ector sup erspace V , for some ǫ 0 , is automatica lly ǫ -rigged for a ll ǫ . Hence, by Theorem 2.18, we automatically get in this ca se a family of r e presen tations of the Ger stenhaber algebra G = S A ( Π g ) on V , dep ending on the par ameter ǫ , which sa tisfies the ǫ - right Leibniz formula (2.12). In par ticula r, in this case, the adjoint representation of G = S A ( Π g ) on itself admits a 1- parameter fa mily of deformations. R emark 2.22. Using the ǫ -right L e ibniz rule (2.12) and r ecalling the relation (2.7) for the par it y in G = S A ( Π g ), o ne ca n find an explicit formula for the Lie deriv ative L X , fo r an arbitrar y element X = X 1 ∧ · · · ∧ X m with X i ∈ g : L X = m X i =1 ( − 1) ¯ p ( X i +1 )+ ··· + ¯ p ( X m )+ m + i ι X 1 · · · ι X i − 1 L X i ι X i +1 · · · ι X m − ǫ X 1 ≤ i

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