Batalin-Vilkovisky structure on Hochschild cohomology with coefficients in the dual algebra
aa r X i v : . [ m a t h . K T ] N ov BATALIN-VILKOVISKY STRUCTURE ON HOCHSCHILDCOHOMOLOGY WITH COEFFICIENTS IN THE DUALALGEBRA
MARCO ANTONIO ARMENTA
Abstract.
We prove that Hochschild cohomology with coefficients in A ∗ = Hom k ( A, k ) together with an A -structural map ψ : A ∗ ⊗ A A ∗ → A ∗ is a Batalin-Vilkovisky algebra. This applies to symmetric, Frobeniusand monomial path algebras. Introduction
Let A be an associative unital algebra projective over a commutativering k . The Hochschild cohomology k -modules of A with coefficients in an A -bimodule M , H • ( A, M ) = M n ≥ H n ( A, M )have been introduced by Hochschild [6] and extensively studied since then.Operations on cohomology have been defined, such as the cup product andthe Gerstenhaber bracket, making it into a Gerstenhaber algebra [4]. Tradlershowed [10] that for symmetric algebras this Gerstenhaber algebra structureon cohomology comes from a Batalin-Vilkovisky operator (BV-operator) andMenichi extended the result [8]. As Tradler mentions, it is important todetermine other families of algebras where this property holds. Lambre-Zhou-Zimmermann proved that this is the case for Frobenius algebras withsemisimple Nakayama automorphism [7]. Independently, Volkov proved withother methods that this holds for Frobenius algebras in which the Nakayamaautomorphism has finite order and the characteristic of the field k does notdivide it [11]. It has also been shown that Calabi-Yau algebras admit theexistence of a BV-operator [5], and that this BV-structure on its cohomologyis isomorphic to the one of the cohomology of the Koszul dual, for a KoszulCalabi-Yau algebra [2]. More generally, for algebras with duality, see [7], aBV-structure is equivalent to a Tamarkin-Tsygan calculus or a differentialcalculus [7]. The proofs of [5], [7] and [10] have in common the use of Connes’differential [3] on homology to define the BV-operator on cohomology.We start by giving an interpretation of Connes’ differential in Hochschildcohomology with coefficients in the A -bimodule A ∗ = Hom k ( A, k ). The useof A ∗ as bimodule of coefficients replaces the inner product which is in force Key words and phrases.
Batalin-Vilkovisky algebras, Hochschild cohomology. for Frobenius algebras [7], [10] as it is shown in Lemma 2.1 and Corollary4.1. For symmetric algebras this induced BV-structure is isomorphic to theone given by Tradler in [10]. In the case of monomial path algebras we givea description of the A -bimodule structure of A ∗ that allows us to constructan A -structural map on A ∗ .To the knowledge of the author there is no other BV -operator entirelyindependent of Connes’ differential.2. Connes’ differential
Connes’ differential is the map B : HH n ( A ) → HH n +1 ( A ) that makesthe Hochschild theory of an algebra into a differential calculus [9]. It is givenby B ([ a ⊗ · · · ⊗ a n ]) = " n X i =0 ( − ni ⊗ a i ⊗ · · · ⊗ a n ⊗ a ⊗ · · · ⊗ a i − . For an A -bimodule M the dual A -bimodule is denoted M ∗ = Hom k ( M, k ).We consider the canonical A -bimodule structure on M ∗ , that is ( af b )( x ) = f ( bxa ) for all a, b ∈ A , all f ∈ M ∗ and all x ∈ M . Let¯ B : H n +1 ( A, A ∗ ) → H n ( A, A ∗ )given by¯ B ([ f ])([ a ⊗ · · · ⊗ a n ])( a ) := n X i =0 ( − ni f ( a i ⊗ · · · ⊗ a n ⊗ a ⊗ · · · ⊗ a i − )(1) . It is straightforward to verify that it is well-defined. Let C : H n ( A, M ∗ ) → H n ( A, M ) ∗ be the morphism C ([ f ])([ x ⊗ a ⊗ · · · ⊗ a n ]) = f ( a ⊗ · · · ⊗ a n )( x ) , for all a i ∈ A , for i = 1 , · · · , n , all x ∈ M and all [ f ] ∈ H n +1 ( A, M ∗ ), see[1]. The evaluation map ev : H n ( A, M ) → H n ( A, M ) ∗∗ can be composedwith the k -dual of C to get a morphism ϕ : H n ( A, M ) → H n ( A, M ∗ ) ∗ which is given by ϕ ([ x ⊗ a ⊗ · · · ⊗ a n ])([ f ]) = f ( a ⊗ · · · ⊗ a n )( x ) . For M = A we obtain a morphism ϕ : HH n ( A ) → H n ( A, A ∗ ) ∗ . The proofof the following lemma is straightforward. ATALIN-VILKOVISKY STRUCTURE 3
Lemma 2.1.
Let k be a commutative ring and let A be an associative andunital k -algebra. The following diagram is commutative HH n ( A ) HH n +1 ( A ) H n ( A, A ∗ ) ∗ H n +1 ( A, A ∗ ) ∗ .B ¯ B ∗ ϕ ϕ If k is a field then ϕ is a monomorphism. If k is a field and HH n ( A ) isfinite dimensional then ϕ : HH n ( A ) → H n ( A, A ∗ ) ∗ is an isomorphism. Batalin-Vilkovisky structure A Gerstenhaber algebra is a triple ( H • , ∪ , [ , ]) such that H • is a graded k -module, ∪ : H n ⊗ H m → H n + m is a graded commutative associative productand [ , ] : H n ⊗ H m → H n + m − is a graded Lie bracket such that it is anti-symmetric [ f, g ] = ( − ( | f |− | g |− [ g, f ], it satisfies the Jacobi identity[ f, [ g, h ]] = [[ f, g ] , h ] + ( − ( | f |− | g |− [ g, [ f, h ]]as well as the Poisson identity[ f, g ∪ h ] = [ f, g ] ∪ h + ( − ( | f |− | g | g ∪ [ f, h ] , for all homogeneus elements f, g, h of H • . We denote by | f | the degree of anhomogeneous element f ∈ H • . A Batalin-Vilkovisky algebra (
BV-algebra )is a Gerstenhaber algebra ( H ∗ , ∪ , [ , ]) together with a morphism∆ : H n +1 → H n such that ∆ = 0 and[ f, g ] = ( − | f | +1 (cid:0) ∆( f ∪ g ) − ∆( f ) ∪ g − ( − | f | f ∪ ∆( g ) (cid:1) . Recall that H ( A, M ) = M A = { m ∈ M | ma = am f or all a ∈ A } for an A -bimodule M . Definition 3.1.
Let M be an A -bimodule. A morphism ψ : M ⊗ A M → M of A -bimodules is called an A -structural map if it is associative, that is ψ ( m ⊗ ψ ( m ⊗ m )) = ψ ( ψ ( m ⊗ m ) ⊗ m ) for all m , m , m ∈ M , and ψ is unital in the sense that there is M ∈ H ( A, M ) such that ψ (1 M ⊗ m ) = ψ ( m ⊗ M ) = m for all m ∈ M . Remark 3.1.
Let ψ : M ⊗ A M → M be an A -structural map. Then the ∪ -product ∪ : H n ( A, M ) ⊗ H m ( A, M ) → H n + m ( A, M ⊗ A M ) can be composed with ψ to obtain ∪ ψ : H n ( A, M ) ⊗ H m ( A, M ) → H n + m ( A, M ) , MARCO ANTONIO ARMENTA that is ( f ∪ ψ g )( a ⊗ · · · ⊗ a n + m ) := ψ (cid:0) f ( a ⊗ · · · ⊗ a n ) ⊗ g ( a n +1 ⊗ · · · ⊗ a n + m ) (cid:1) . Our assumptions on ψ imply that H • ( A, M ) is an associative and unital k -algebra. We will denote H • ψ ( A, M ) the k -algebra H • ( A, M ) endowed with the ∪ ψ -product. In case M = A ∗ , we have the following. Lemma 3.1.
Let A be an associative unital k -algebra and let ψ : A ∗ ⊗ A A ∗ → A ∗ be an A -structural map. Then H • ψ ( A, A ∗ ) is a Gerstenhaber algebra.Proof. Let d ∗ be the differential on the complex that calculates H • ( A, A ∗ )and let f, g ∈ H • ( A, A ∗ ) be homogeneous elements. The following relationis well known, see [4], f ∪ g − ( − | f || g | g ∪ f = d ∗ ( g )¯ ◦ f + ( − | f | d ∗ ( g ¯ ◦ f ) + ( − | f |− g ¯ ◦ d ∗ ( f ) , where g ¯ ◦ f ( a ⊗ · · · ⊗ a | f | + | g |− ) is by definition | g | X i =0 ( − j g ( a ⊗ · · · a i − ⊗ f ( a i ⊗ · · · ⊗ a i + | f |− ) ⊗ a i + | f | ⊗ · · · ⊗ a | f | + | g |− ) , for j = ( i − | f | − f and g are cocycles, we get that the cup productis graded commutative and since ψ is k -linear we get that the ∪ ψ -product isgraded commutative. Define the bracket in terms of ¯ B and the ∪ ψ -productas [ f, g ] ψ := ( − ( | f |− | g | (cid:0) ¯ B ( f ∪ ψ g ) − ¯ B ( f ) ∪ ψ g − ( − | f | f ∪ ψ ¯ B ( g ) (cid:1) . Hence the graded k -module H • ψ ( A, A ∗ ) with the ∪ ψ -product and the bracket[ , ] ψ is a Gerstenhaber algebra. (cid:3) Theorem 3.1.
Let A be an associative unital k -algebra and let ψ : A ∗ ⊗ A A ∗ → A ∗ be an A -structural map. Then the data (cid:16) H • ψ ( A, A ∗ ) , ∪ ψ , [ , ] ψ , ¯ B (cid:17) is a BV-algebra.Proof. Since the following diagram is commutative HH n ( A ) HH n +1 ( A ) HH n +2 ( A ) H n ( A, A ∗ ) ∗ H n +1 ( A, A ∗ ) ∗ H n +2 ( A, A ∗ ) ∗ .B B ¯ B ∗ ¯ B ∗ ϕ ϕ ϕ we have that ¯ B = 0. Then H • ψ ( A, A ∗ ) is a BV-algebra with the bracketdefined as in the last lemma. (cid:3) ATALIN-VILKOVISKY STRUCTURE 5 Frobenius and Symmetric algebras
Assume that A is a symmetric algebra, i.e. a finite dimensional algebrawith a symmetric, associative and non-degenerate bilinear form <, > : A ⊗ A → k , where associative means < ab, c > = < a, bc > for all a, b, c ∈ A . The bilinear form defines an isomorphism of A -bimodules Z : A → A ∗ given by Z ( a ) = < a, − > . It is shown in [10] that this defines a BV -operator on Hochschild cohomology, where ∆ f is defined such that for f ∈ HH n ( A ) we have < ∆ f ( a ⊗· · ·⊗ a n − ) , a n > = n X i =1 ( − i ( n − < f ( a i ⊗· · · a n ⊗ a · · ·⊗ a i − ) , > . Corollary 4.1. If A is a symmetric algebra, then there is an A -structuralmap ψ : A ∗ ⊗ A A ∗ → A ∗ such that the BV-algebras HH • ( A ) and H • ψ ( A, A ∗ ) are isomorphic.Proof. Let Z : A → A ∗ be the isomorphism of A -bimodules given by thebilinear form of A . We will denote Z ∗ : HH • ( A ) → H • ψ ( A, A ∗ ) the iso-morphism induced by composition with Z . Then the following diagram iscommutative HH n ( A ) HH n − ( A ) H n ( A, A ∗ ) H n − ( A, A ∗ ) . ∆¯ BZ ∗ Z ∗ Indeed,( ¯ B ◦ Z ∗ )([ f ])( a ⊗ · · · ⊗ a n − )( a )= ¯ B ( Z ◦ f )( a ⊗ · · · ⊗ a n − )( a )= P n − i =0 ( − ( n − i ( Z ◦ f )( a i ⊗ · · · ⊗ a n − ⊗ a ⊗ · · · ⊗ a i − )(1)= Z ◦ (cid:16)P n − i =0 ( − ( n − i f ( a i ⊗ · · · ⊗ a n − ⊗ a ⊗ · · · ⊗ a i − )(1) (cid:17) = ( Z ∗ ◦ ∆)([ f ])( a ⊗ · · · ⊗ a n − )( a ) . Using the isomorphism given by the product A ⊗ A A ∼ = A the transport ofthe algebra structure of A to A ∗ via Z gives the A -structural map ψ = Z ◦ ( Z ⊗ Z ) − : A ∗ ⊗ A A ∗ → A ∗ . This isomorphism satisfies the associativity and unity conditions of remark3.1, since the product of A is associative and has a unit. Even more, there MARCO ANTONIO ARMENTA are commutative diagrams where the vertical maps are isomorphisms HH n ( A ) ⊗ HH m ( A ) HH n + m ( A ) H n ( A, A ∗ ) ⊗ H m ( A, A ∗ ) H n + m ( A, A ∗ ) , ∪∪ ψ Z ∗ ⊗ Z ∗ Z ∗ HH n ( A ) ⊗ HH m ( A ) HH n + m − ( A ) H n ( A, A ∗ ) ⊗ H m ( A, A ∗ ) H n + m − ( A, A ∗ ) . [ , ][ , ] ψ Z ∗ ⊗ Z ∗ Z ∗ Indeed, Z ∗ ( f ) ∪ ψ Z ∗ ( g ) = ψ ◦ ( Z ⊗ Z )( f ∪ g )= Z ◦ ( Z ⊗ Z ) − ◦ ( Z ⊗ Z )( f ∪ g )= Z ◦ ( f ∪ g )= Z ∗ ( f ∪ g ) , and[ Z ∗ f, Z ∗ g ] ψ = ( − ( | f |− | g | (cid:16) ¯ b (cid:0) Z ∗ f ∪ ψ Z ∗ g (cid:1) − ¯ b ( Z ∗ f ) ∪ ψ Z ∗ g − ( − | f | Z ∗ f ∪ ψ ¯ b ( Z ∗ g ) (cid:17) = ( − ( | f |− | g | (cid:16) ¯ b (cid:0) Z ∗ ( f ∪ ψ g ) (cid:1) − Z ∗ (∆ f ) ∪ ψ Z ∗ g − ( − | f | Z ∗ f ∪ ψ Z ∗ (∆) g (cid:17) = ( − ( | f |− | g | (cid:16) Z ∗ ∆( f ∪ g ) − Z ∗ (∆ f ∪ g ) − ( − | f | Z ∗ ( f ∪ ∆ g ) (cid:17) = ( − ( | f |− | g | Z ∗ (cid:16) ∆( f ∪ g ) − (∆ f ∪ g ) − ( − | f | ( f ∪ ∆ g ) (cid:17) = Z ∗ [ f, g ] . Commutativity of these diagrams implies that the BV -algebras HH • ( A )and H • ψ ( A, A ∗ ) are isomorphic. (cid:3) Remark 4.1.
Observe that choosing ∆ := ( Z ∗ ) − ¯ bZ ∗ gives HH • ( A ) thestructure of a BV-algebra. Assume now that A is a Frobenius algebra, i.e. a finite dimensional alge-bra with a non-degenerate associative bilinear form < − , − > : A × A → k .For every a ∈ A there exist a unique N ( a ) ∈ A such that < a, − > = < − , N ( a ) > . The map N : A → A turns out to be an algebra isomorphismand is called the Nakayama automorphism of the Frobenius algebra A . Fol-lowing [7] we consider the A -bimodule A N whose underlying k -module is A and the corresponding actions are axb = ax N ( b ) . Hence the morphism Z : A N → A ∗ given by Z ( a ) = < a, − > is an isomor-phism of A -bimodules [7]. The morphism µ : A N ⊗ A A N → A N ATALIN-VILKOVISKY STRUCTURE 7 given by µ ( a ⊗ b ) = a N ( b ) is a morphism of A -bimodules since µ ( ab ⊗ A cd ) = ab N ( cd ) = ab N ( c ) N ( d ) = ab N ( c ) d = aµ ( b ⊗ A c ) d and it is well-defined since µ ( ac ⊗ b ) = µ ( a N ( c ) ⊗ b ) = a N ( c ) N ( b ) = a N ( cb ) = µ ( a ⊗ cb )for all a, b, c, d ∈ A N . It is also unital and associative since N (1) = 1, and µ (cid:0) µ ( a ⊗ b ) ⊗ c (cid:1) = µ (cid:0) a N ( b ) ⊗ c (cid:1) = µ (cid:0) a ⊗ bc (cid:1) = µ (cid:0) a ⊗ b N ( c ) (cid:1) = µ (cid:0) a ⊗ µ ( b ⊗ c ) (cid:1) . Then ψ = Z ◦ µ ◦ ( Z ⊗ A Z ) − : A ∗ ⊗ A A ∗ → A ∗ is an A -structural map. Corollary 4.2.
Let A be a Frobenius algebra with diagonalizable Nakayamaautomorphism, then the BV-algebras HH • ψ ( A, A ∗ ) and HH • ( A, A N ) are iso-morphic.Proof. Hochschild cohomology of A with coefficients in A N is isomorphic, see[7], to Hochschild cohomology of A with coefficients in A N corresponding tothe eigenvalue 1 ∈ k of the linear transformation N , HH • ( A, A N ) ∼ = HH • ( A, A N ) . The BV-operator of HH • ( A, A N ) is the transpose of Connes’ differential B N ([ a ⊗ · · · ⊗ a n ]) = " n X i =0 ( − in a i ⊗ · · · ⊗ a n ⊗ N ( a ) ⊗ · · · ⊗ N ( a i − ) , with respect to the duality given in [7]. By finite dimensionality arguments,this morphism turns out to be the k -dual of ϕ , namely ∂ : HH • ( A, A ∗ ) ∗ → HH • ( A ) . The compatibility conditions for the ∪ -product and the Gerstenhaber bracketare proved similarly. (cid:3) Monomial path algebras
Let Q be a finite quiver with n vertices and consider a monomial pathalgebra A = kQ/ h T i , that is, T is a subset of paths in Q of length greateror equal than 2. We do not require the algebra A to be finite dimensional.We write s ( ω ) and t ( ω ) for the source and the target of ω . A basis P of A isgiven the set of paths of Q which do not contain paths of T . Let P ∨ be thedual basis of P , and for ω ∈ P we denote ω ∨ its dual. Let α ∈ P and define ω /α as the subpath of ω that starts in s ( ω ) and ends in s ( α ) if α is a subpathof ω such that t ( α ) = t ( ω ), and zero otherwise. Let β ∈ P and define β \ ω as the subpath of ω that starts at t ( β ) and ends in t ( ω ) if β is a subpathof ω such that s ( β ) = s ( ω ), and zero otherwise. The canonical A -bimodule MARCO ANTONIO ARMENTA structure of A ∗ is isomorphic to the one given by linearly extending thefollowing action α.ω ∨ .β = ( β \ ω /α ) ∨ . Now we construct an A -structural map for A ∗ . For ω, γ ∈ P we define ω ∨ · γ ∨ = (cid:20) ( γω ) ∨ if t ( ω ) = s ( γ )0 otherwise (cid:21) and extend by linearity. Observe that γ β \ ω = γ /β ω , then( ω ∨ .β ) · γ ∨ = ( β \ ω ) ∨ · γ ∨ = ( γ β \ ω ) ∨ = ( γ /β ω ) ∨ = ω ∨ · ( γ /β ) ∨ = ω ∨ · ( β.γ ∨ ) . Therefore, by linearly extending ψ ( ω ∨ ⊗ γ ∨ ) = ω ∨ · γ ∨ we get a morphismof k -modules ψ : A ∗ ⊗ A A ∗ → A ∗ . It is a morphism of A -bimodules since α. ( ω ∨ · γ ∨ ) .β = α. ( γω ) ∨ .β = ( β \ γω /α ) ∨ = ( ω /α ) ∨ · ( β \ γ ) ∨ = ( α.ω ∨ ) · ( γ ∨ .β ) . The morphism ψ is associative since the product of A is associative. Let e , ..., e n be the idempotents of A given by the vertices of Q . Define 1 ∗ = e ∨ + · · · + e ∨ n and observe that if α is a basic element of A of length greateror equal than one then1 ∗ .α = e ∨ .α + · · · + e ∨ n .α = 0 = α.e ∨ + · · · + α.e ∨ n = α. ∗ for every i = 1 , ..., n . Moreover,1 ∗ .e i = e ∨ .e i + · · · + e ∨ n .e i = e ∨ i = e i .e ∨ + · · · + e i .e ∨ n = e i . ∗ so we get that 1 ∗ ∈ H ( A, A ∗ ). Finally,1 ∗ · ω ∨ = e ∨ · ω ∨ + · · · + e ∨ n · ω ∨ = e ∨ t ( ω ) · ω ∨ = ω ∨ and analogously ω ∨ · ∗ = ω ∨ . Therefore ψ is an A -structural map. Corollary 5.1.
Let A be a monomial path algebra. Then H • ψ ( A, A ∗ ) is aBV-algebra. Acknowledgements.
I want to thank Claude Cibils and Ricardo Camposfor some very useful talks at IMAG, University of Montpellier, during theRecontre 2018 du GdR de Topologie Alg´ebrique. The author received fundsby CIMAT, CNRS, CONACyT and EDUCAFIN.
References
1. H. Cartan and S. Eilenberg,
Homological algebra , Princeton Landmarks in Mathemat-ics, Princeton University Press, Princeton, NJ, 1999, With an appendix by David A.Buchsbaum, Reprint of the 1956 original.2. X. Chen, S. Yang, and G. Zhou,
Batalin-Vilkovisky algebras and the noncommutativePoincar´e duality for Koszul Calabi-Yau algebras , J. Pure Appl. Algebra (2016),2500–2532.3. A. Connes,
Non-commutative differential geometry , Publ. Math. IH´ES (1985), 257–300. ATALIN-VILKOVISKY STRUCTURE 9
4. M. Gerstenhaber,
The cohomology structure of an associative ring , Ann. of Math. (1963), 267–288.5. V. Ginzburg, Calabi-Yau algebras , arXiv:math/0612139.6. G. Hochschild,
On the cohomology groups of an associative algebra , Ann. Math. (2) (1945), 58–67.7. T. Lambre, G. Zhou, and A. Zimmermann, The Hochschild cohomology ring of aFrobenius algebra with semisimple Nakayama automorphism is a Batalin-Vilkoviskyalgebra , J. Algebra (2016), no. 15, 103–131.8. L. Menichi,
Batalin-Vilkovisky algebra structure on Hochschild cohomology , Bulletinde la SMF (2009), no. 2, 277–295.9. D. Tamarkin and B. Tsygan,
Noncommutative differential calculus, homotopy BValgebras and formality conjectures , Methods Funct. Anal. Topology (2000), no. 2,85–100.10. T. Tradler, The Batalin-Vilkovisky algebra on Hochschild cohomology induced by in-finity inner products , Ann. Inst. Fourier (2008), no. 7, 2351–2379.11. Y. Volkov, BV-differential on Hochschild cohomology of Frobenius algebras , J. Pure.Appl. Algebra (2016), no. 10, 3384–3402.
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