Noncommutative Differential Calculus Structure on Secondary Hochschild (co)homology
aa r X i v : . [ m a t h . R A ] M a r NONCOMMUTATIVE DIFFERENTIAL CALCULUS STRUCTURE ONSECONDARY HOCHSCHILD (CO)HOMOLOGY
APURBA DAS, SATYENDRA KUMAR MISHRA, AND ANITA NAOLEKAR
Abstract.
Let B be a commutative algebra and A be a B -algebra (determined by an algebrahomomorphism ε : B → A ). M. D. Staic introduced a Hochschild like cohomology H • (( A, B, ε ); A )called secondary Hochschild cohomology, to describe the non-trivial B -algebra deformations of A .J. Laubacher et al later obtained a natural construction of a new chain (and cochain) complex C • ( A, B, ε ) (resp. C • ( A, B, ε )) in the process of introducing the secondary cyclic (co)homology.It turns out that unlike the classical case of associative algebras (over a field), there exist different(co)chain complexes for the B -algebra A . In this paper, we establish a connection between the two(co)homology theories for B -algebra A . We show that the pair (cid:0) H • (( A, B, ε ); A ) , HH • ( A, B, ε ) (cid:1) forms a noncommutative differential calculus, where HH • ( A, B, ε ) denotes the homology of thecomplex C • ( A, B, ε ). Introduction
Hochschild cohomology was introduced by G. Hochschild to study certain extensions of associativealgebras. In 1963, a pioneer work of M. Gerstenhaber related Hochschild cohomology with thedeformations of associative algebras [3]. He further proved that the Hochschild cohomology of anassociative algebra carries a rich algebraic structure which is now known as
Gerstenhaber algebra (see [3] for details). These structures appear in the context of the exterior algebra of Lie algebras,multivector fields on smooth manifolds, and differential forms on Poisson manifolds. A Gerstenhaberalgebra is a graded commutative associative algebra ( A = ⊕ i ∈ Z A i , ∪ ) together with a degree − − , − ] on A satisfying the following compatibility condition.[ a, b ⌣ c ] = [ a, b ] ⌣ c + ( − ( | a |− | b | b ⌣ [ a, c ] . A Batalin-Vilkovisky (BV-) operator on a Gerstenhaber algebra ( A = ⊕ i ∈ Z A i , ⌣ ) is a square zerodegree − △ : A → A with[ a, b ] = ± ( △ ( a ⌣ b ) − ( △ a ) ⌣ b − ( − | a | a ⌣ ( △ b )) . This shows that the bracket [ − , − ] obstructs △ to be a derivation with respect to the product ⌣ .A Gerstenhaber algebra with a BV-operator is called a Batalin-Vilkovisky algebra (BV-algebra).Let A is an associative k -algebra, B a commutative k -algebra and ε : B → A an algebra morphismwith ε ( B ) ⊂ Z ( A ), the center of A . In 2016, M. Staic [12,13] introduced the secondary Hochschildcohomology H • (( A, B, ε ); M ) of the triple ( A, B, ε ) with coefficients in a B -symmetric A -bimodule M . This cohomology, motivated by an algebraic version of the second Postnikov invariant [12],controls the deformations of the B -algebra structures on A [[ t ]]. When B is the underlying field k , thesecondary Hochschild cohomology coincides with the classical Hochschild cohomology. Several resultswhich are true for classical Hochschild cohomology theory for an associative algebra with coefficientsin itself have analogues for the secondary Hochschild cohomology H • (( A, B, ε ) , A ) of a triple ( A, B, ε )with coefficients in A . In particular, it is proved in [14] that the secondary complex C • (( A, B, ε ); A )is a multiplicative non-symmetric operad, which induces a natural homotopy Gerstenhaber algebrastructure on it. Consequently, one obtains a natural Gerstenhaber algebra structure on the secondaryHochschild cohomology H • (( A, B, ε ); A ). Corresponding Author [A2] email: [email protected]
Mathematics Subject Classification.
Key words and phrases.
Secondary Hochschild (co)homology; Gerstenhaber algebra; comp module; noncommutativedifferential calculus.
The simplicial structure of the complex C • (( A, B, ε ); M ) that defines the secondary cohomology H • (( A, B, ε ); M ) is discussed in [8]. For this purpose, the authors introduce a simplicial module B ( A, B, ε ) over a certain simplicial algebra, and call it the secondary bar simplicial module . Thissimplicial module B ( A, B, ε ) is the analogue of the bar resolution associated with a k-algebra A,in Hochschild cohomology. Using this simplicial module, the cohomology H • (( A, B, ε ); M ) of thetriple ( A, B, ε ) with coefficients in M can be realized as the homology of the associated complexof co-simplicial module. This construction of the secondary bar simplicial module leads to naturalconstructions of the secondary Hochschild homology groups H • (( A, B, ε ) , M ) of the triple ( A, B, ε )with coefficients in M . Also the authors in [8] have given natural constructions of secondary coho-mology groups HH • ( A, B, ε ) and the secondary cyclic cohomology groups HC • ( A, B, ε ) associatedto a triple (
A, B, ε ). This has also prompted the authors to define the corresponding homologygroups HH • ( A, B, ε ) and HC • ( A, B, ε ) associated to the triple (
A, B, ε ). We refer to [8] and [1] formore Hochschild-like results on the secondary Hochschild and secondary cyclic (co)homology groups,associated to the triple (
A, B, ε ).This article aims to establish a connection between the two (co)homology theories for a triple(
A, B, ε ) introduced in [13] and [8]. We show that the pair (cid:0) H • (( A, B, ε ); A ) , HH • ( A, B, ε ) (cid:1) formsa noncommutative differential calculus. A calculus ( A , Ω) consists of a Gerstenhaber algebra A anda graded space Ω such that • Ω carries a Gerstenhaber module structure over A , and • there exists a differential B : Ω • → Ω • +1 satisfying the Cartan-Rinehart homotopy formula(see Definition 2.4).In differential geometry and noncommutative geometry, there are several concrete examples of non-commutative differential calculi [2,11,15]. These examples include the classical calculus of multi-vector fields and differential forms, a calculus on the pair Hochschild cohomology and Hochschildhomology of associative algebras, the calculus for Poisson structures, and the calculus for general leftHopf algebroids with respect to general coefficients (see Section 6, [7]). In [7] N. Kowalzig provedthat if a comp module structure [6] is cyclic over a multiplicative operad, then this cyclic compmodule structure induces a noncommutative differential calculus (in the sense of [11,15]) on the pairof the associated homology of the cyclic k -module and the cohomology of the operad.In this article, we consider the multiplicative non-symmetric operad structure on secondarycochain complex C • (( A, B, ε ); A ) from [14]. The secondary Hochschild chain complex associatedto a triple ( A, B, ε ) is denoted by C • ( A, B, ε ). We define comp module actions of the operad C • (( A, B, ε ); A ) on the complex C • ( A, B, ε ). With these actions, we prove that the complex C • ( A, B, ε ) is a cyclic comp module over the operad C • (( A, B, ε ); A ). Subsequently, it followsthat the pair of underlying homologies (cid:0) H • (( A, B, ε ); A ) , HH • ( A, B, ε ) (cid:1) forms a (noncommutative)differential calculus.In Section 2, we recall the definitions and results related to multiplicative non-symmetric oper-ads and cyclic unital comp module structures over multiplicative operads. In Section 3, we recallsecondary Hochschild (co)homology for a triple ( A, B, ε ). In particular, we recall the definitions ofthe complexes C • (( A, B, ε ); A ) and C • ( A, B, ε ) from [8,13]. We also recall the multiplicative operadstructure on the complex C • (( A, B, ε ); A ) from [14]. In Section 4, we define comp module actionsof C • (( A, B, ε ); A ) on the complex C • ( A, B, ε ) and obtain a comp module structure on C • ( A, B, ε ).We further consider an operator t : C • ( A, B, ε ) → C • ( A, B, ε ) to show that the comp module struc-ture is cyclic. Finally, we conclude that there exists a (noncommutative) differential calculus on thepair (cid:0) H • (( A, B, ε ); A ) , HH • ( A, B, ε ) (cid:1) . This provides a connection between the cohomology groupsof the triple ( A, B, ε ) and the homology groups associated to the triple (
A, B, ε ). ALCULUS STRUCTURE ON SECONDARY HOCHSCHILD (CO)HOMOLOGY 3 Comp modules over (non-symmetric) operads
In this section, we recall the notion of cyclic comp modules over operads and the related resultsfrom [7]. In particular, we recall that a cyclic comp module structure over a multiplicative operadinduces a noncommutative differential calculus.2.1.
Definition.
A non-symmetric (unital) operad O in the category of k -modules is a sequence of k -modules {O n } n ≥ equipped with k -linear maps ◦ i : O n ⊗ O m → O n + m − , for m, n ≥ ≤ i ≤ n, and an element ∈ O such that the following identities hold f ◦ i g = 0 if n < i or n = 0 , ( f ◦ i g ) ◦ j h = ( f ◦ j h ) ◦ i + p − g if j < i,f ◦ i ( g ◦ j − i +1 h ) if i ≤ j < m + i, ( f ◦ j − m +1 h ) ◦ i f if j ≥ m + i, and f ◦ i = f = ◦ f, for all i ≤ n, where f ∈ O n , g ∈ O m , and h ∈ O p .If O is an operad, then one can define a circle product ◦ : O n ⊗ O m → O n + m − by f ◦ g = n X i =1 ( − ( i − m − f ◦ i g, for f ∈ O n and g ∈ O m . Subsequently, a degree − f, g ] = f ◦ g − ( − ( n − m − g ◦ f, for f ∈ O n and g ∈ O m . We call the operad O , a multiplicative operad if there exists an element µ ∈ O and an element e ∈ O such that µ ◦ µ = µ ◦ µ and µ ◦ e = = µ ◦ e . The multiplication µ on the operad O induces a differential δ µ : O n → O n +1 , given by δ µ ( f ) = [ µ, f ] for f ∈ O n . Let us denote by H • µ ( O ),the cohomology space of the complex ( O , δ µ ). The multiplication µ also induces a cup product ⌣ : O n ⊗ O m → O n + m on the complex O , which is given by f ⌣ g = ( µ ◦ f ) ◦ g, for f ∈ O m and g ∈ O n . The graded commutative product ⌣ and the Gerstenhaber bracket [ , ] induce a Gerstenhaberalgebra structure on the cohomology H • µ ( O ) (see [4] for more details).2.2. Definition.
A unital (left) comp module M • over an operad O • := {O n } n ≥ is a sequence of k -modules {M n } n ≥ equipped with comp module maps • i : O m ⊗M p → M p − m +1 , for 1 ≤ i ≤ p − m +1and 0 ≤ m ≤ p such that(1) f • i ( g • j x ) = g • j ( f • i + n − x ) if j < i, ( f ◦ j − i +1 g ) • i x if j − m < i ≤ j,g • j − m +1 ( f • i x ) if 1 ≤ i ≤ j − m, (2) • i x = x for 1 ≤ i ≤ p, where f ∈ O m , g ∈ O n , and x ∈ M p , for m ≥ , n, p ≥
0. The maps • i are zero, for m > p .2.3. Definition.
A unital comp module M • is called para-cyclic if it is additionally equipped with(i) an extra comp module maps • : O m ⊗ M p → M p − m +1 , for 0 ≤ m ≤ p + 1 (the maps • areassumed to be zero, for m > p + 1) such that the relations (1)-(2) also hold true for i = 0, and(ii) a k -linear map t : M p → M p , for p ≥ , such that t ( f • i x ) = f • i +1 t ( x ) , APURBA DAS, SATYENDRA KUMAR MISHRA, AND ANITA NAOLEKAR for f ∈ O m , x ∈ M p , and 0 ≤ i ≤ p − m . If the map t : M p → M p satisfies the condition: t p +1 = Id , then the para-cyclic comp module M • is said to be ‘ cyclic ’ comp module over theoperad O • .Let ( O • , µ ) be a multiplicative non-symmetric operad and M • be a cyclic comp module over O • .Then let us recall from [7] that there is a cyclic k -module structure on M • with cyclic operator t : M p → M p , face maps d i : M p → M p − and degeneracies s j : M p → M p +1 defined as follows d i ( x ) = µ • i x, for i = 0 , . . . , p − ,d p ( x ) = µ • t ( x ) ,s j ( x ) = • j +1 x, for j = 0 , . . . , p, where x ∈ M p . Thus, a simplicial boundary b , a norm operator N , an extra degeneracy s − , andthe cyclic differential B are defined as follows b := p X i =0 ( − i d i , N := p X i =0 ( − ip t i , s − := t s p , B := ( id − t ) s − N. The normalised complex N ( M ) • is the quotient of the complex M • by the subcomplex spannedby the images of degeneracy maps s j , for j = 0 , , . . . , p . The map B on the normalised complex N ( M ) • is given by(3) B ( x ) = s − N ( x ) = p X i =0 ( − ip • t i ( x ) , for x ∈ N ( M ) p . The noncommutative differential calculus associated to a cyclic comp module overa multiplicative operad.
Definition ([7]) . A graded k -module Ω := ⊕ Ω n is called a Gerstenhaber module over Ger-stenhaber algebra A if there exist maps i : A m ⊗ Ω p → Ω p − m , and L : A m ⊗ Ω p → Ω p − m +1 suchthat(i) the action i makes Ω a graded module over the graded commutative associative algebra ( A , ∧ );(ii) the action L makes Ω a graded Lie module over the graded Lie algebra ( A [1] , [ − , − ]);(iii) for any X ∈ A m and Y ∈ A n +1 , the following relation holds i [ X,Y ] = i X L Y − ( − mn L Y i X . The Gerstenhaber module Ω = ⊕ Ω n is called a Batalin-Vilkovisky module if there exists a k -linearmap B : Ω n → Ω n +1 such that B = 0, and it satisfies the following Cartan-Rinehart homotopyformula L X = B ◦ i X − ( − m i X ◦ B, for X ∈ A m . A pair ( A , Ω), where A is a Gerstenhaber algebra and Ω is a Batalin-Vilkovisky module over A , iscalled a noncommutative differential calculus .Let ( O , µ ) be a multiplicative operad and M • be a cyclic unital comp module over ( O , µ ).2.5. Definition ([7]) . For f ∈ O m , the cap product and the Lie derivative are given as follows.(i) The cap product i f : M p → M p − m is defined by i f x = ( µ ◦ f ) • x, for x ∈ M p . (ii) The Lie derivative L f : M p → M p − m +1 of x ∈ M p along an element f ∈ O m is defined by L f ( x ) = n − p +1 P i =1 ( − ( p − i − f • i x + p P i =1 ( − n ( i − p − f • t ( i − ( x ) if m < p + 1 , ( − p − f • N ( x ) if m = p + 1 , m > p + 1 . ALCULUS STRUCTURE ON SECONDARY HOCHSCHILD (CO)HOMOLOGY 5
It is shown in [7] that the cap product and the Lie derivative satisfy the following identities.(4) i δ µ f = b ◦ i f − ( − m i f ◦ b , [ b , L f ] + L δ µ f = 0 . (5) i f i g = i f⌣g , [ L f , L g ] = L [ f,g ] . From equation (4), it follows that the cap product and Lie derivative descend to well-defined oper-ators on the homology H • ( M • ). Moreover, from equation (5), the cap product makes H • ( M • ) agraded module over the algebra ( H • ( O • ) , ⌣ ), and the Lie derivative makes H • ( M • ) a graded Liemodule over the graded Lie algebra ( H • +1 ( O • ) , [ , ]). For any two cocycles f ∈ O m and g ∈ O n ,the induced operators L f : H • ( M • ) → H •− m +1 ( M • ) and i g : H • ( M • ) → H •− n ( M • )satisfy the relation(6) [ i f , L g ] = i [ f,g ] . The identity (6) shows that the homology H • ( M ) is a Gerstenhaber module over the Gerstenhaberalgebra H • µ ( O ).In fact, this Gerstenhaber module structure extends to a Batalin-Vilkovisky module structure.Let us consider the normalised complexes N ( M ) • and N ( O ) • . Also, recall that (co)homology ofthe normalised (co)chain complex is the same as the (co)homology of the original complex. Theinduced norm operator on the normalised complex N ( M ) • induces a well defined k -linear map B : H • ( M ) → H • +1 ( M ) satisfying B = 0. For any m -cocycle f ∈ N ( O ) m , the operators L f : H • ( M ) → H •− m +1 ( M ) and i f : H • ( M ) → H •− m ( M )satisfy the following Cartan-Rinehart homotopy formula L f = [ B, i f ] := B ◦ i f − ( − m i f ◦ B. i.e., H • ( M ) is a Batalin-Vilkovisky module over the Gerstenhaber algebra H • µ ( O ). Thus, the pair (cid:0) H • µ ( O ) , H • ( M ) (cid:1) forms a noncommutative differential calculus (see [7] for more details).3. Secondary Hochschild (co)homology
Let A be an associative k -algebra and B be a commutative k -algebra. Suppose that there is a k -algebra morphism ε : B → A such that ε ( B ) ⊂ Z ( A ) the center of A . We denote the above structureby a triple ( A, B, ε ). In this section, we recall the notion of secondary Hochschild (co)homology fora triple (
A, B, ε ) introduced in [8,13,14].3.1.
Secondary Hochschild (co)homology of the triple ( A, B, ε ) with coefficients in A . Let (
A, B, ε ) be a triple. With the above notations, the secondary Hochschild cohomology of atriple (
A, B, ε ) with coefficients in A is given by considering the cochain complex C • (( A, B, ε ); A ) = ⊕ n ≥ C n (( A, B, ε ); A ), where C n (( A, B, ε ); A ) := Hom k ( A ⊗ n ⊗ B ⊗ n ( n − , A ) , for n ≥ , and the differential δ ε : C n − (( A, B, ε ); A ) → C n (( A, B, ε ); A ) is defined by δ ε ( f ) ⊗ a b , · · · b ,n − b ,n a · · · b ,n − b ,n · · · · · · · · · · a n − b n − ,n · · · a n := a ε ( b , · · · b ,n ) f ⊗ a b , · · · a ,n a · · · b ,n · · · · · · · · · a n + APURBA DAS, SATYENDRA KUMAR MISHRA, AND ANITA NAOLEKAR n − X i =1 ( − i f ⊗ a b , · · · b ,i b ,i +1 · · · b ,n − b ,n a · · · b ,i b ,i +1 · · · b ,n − b ,n · · · · · · · · · · · · · · a i a i +1 ε ( b i,i +1 ) · · · b i,n − b i +1 ,n − b i,n b i +1 ,n · · · · · · · · · · · · · · · · · a n − b n − ,n · · · · · · a n +( − n f ⊗ a b , · · · a ,n − a · · · b ,n − · · · · · · · · · a n − ε ( b ,n · · · b n − ,n ) a n . The cohomology of this cochain complex is denoted by H • (( A, B, ε ); A ) and it is called thesecondary Hochschild cohomology of the triple ( A, B, ε ) with coefficients in A .In the present article, we do not use the notion of secondary Hochschild homology H • (( A, B, ε ); A )of the triple ( A, B, ε ) with coefficients in A . So, we skip the definition of H • (( A, B, ε ); A ) (see [12]for details).3.1. Remark. If B = k and ε : k → A defining the k -algebra structure on A , the secondaryHochschild (co)homology coincides with the classical Hochschild (co)homology of the associativealgebra A .3.2. Gerstenhaber algebra structure on the secondary Hochschild cohomology.
Let usconsider an operad {O n } n ≥ , where O n = C n (( A, B, ε ); A ) . The underlying k -bilinear maps ◦ i : C n (( A, B, ε ); A ) ⊗ C m (( A, B, ε ); A ) → C n + m − (( A, B, ε ); A ) , for n, m ≥ ≤ i ≤ n, are given as follows( f ◦ i g )( T n + m − )(7) = f ⊗ a · · · b ,i − m + i − Q j = i b ,j b ,m + i · · · b ,n + m − · · · b ,i − m + i − Q j = i b ,j b ,m + i · · · b ,n + m − · · · · · · · · · · a i − m + i − Q j = i b i − ,j b i − ,m + i · · · b i − ,n + m − · · · g ( T im + i − ) m + i − Q j = i b j,m + i · · · m + i − Q j = i b j,n + m − · · · a m + i · · · b m + i,n + m − · · · · · · · · · · · · · a n + m − , where T ik := ⊗ a i b i, · · · b i,k − b i,k a i +1 · · · b i +1 ,k − b i +1 ,k · · · · · · · · · · a k − b k − ,k · · · a k ALCULUS STRUCTURE ON SECONDARY HOCHSCHILD (CO)HOMOLOGY 7
Moreover, consider an element µ ∈ O = C (( A, B, ε ); A ) given by(8) µ ⊗ a b a !! = ε ( b ) a a . Then, it follows that {O n } n ≥ is a multiplicative non-symmetric operad with the multiplication µ ∈ O and 1 ∈ O = A (see [14] for more details).The precise construction of Gerstenhaber algebra structure on H • (( A, B, ε ); A ) is described asfollows: the pre-Lie bracket ◦ is given by f ◦ g = n X i =1 ( − ( i − m − f ◦ i g, for f ∈ C n , g ∈ C m ;the graded Lie algebra structure on ⊕ n ≥ C n (( A, B, ε ); A ) is given by[ f, g ] = f ◦ g − ( − ( n − m − g ◦ f, for f ∈ C n , g ∈ C m ;the cup-product on C • (( A, B, ε ); A ) is defined by( f ⌣ g ) ⊗ a b , · · · b ,n + m a · · · b ,n + m · · · · · · · a m + n = m Y i =1 m + n Y j = m +1 ε ( b i,j ) g ( T m ) f ( T m +1 m + n ) . The induced operations on the secondary Hochschild cohomology H • (( A, B, ε ); A ) make it a Ger-stenhaber algebra.3.3. Secondary Hochschild (co)homology associated to the triple ( A, B, ε ) . We recall thenotion of secondary Hochschild homology associated to the triple (
A, B, ε ) from [12]. Let us considerthe chain complex (cid:0) C • ( A, B, ε ) := ⊕ n ≥ C n ( A, B, ε ) , ∂ (cid:1) , where C n ( A, B, ε ) := A ⊗ ( n +1) ⊗ B n ( n +1)2 , n ≥ ∂ : C n → C n − is defined by ∂ ⊗ a b , · · · b ,n − b ,n a · · · b ,n − b ,n · · · · · · · · a n − b n − ,n · · · a n := n − X i =0 ( − i ⊗ a · · · b ,i − b ,i b ,i +1 b ,i +2 · · · b ,n · · · · · · · · · · a i − b i − ,i b i − ,i +1 b i − b i +2 · · · b i − ,n · · · ε ( b i,i +1 ) a i a i +1 b i,i +2 b i +1 ,i +2 · · · b i,n b i +1 ,n · · · a i +2 · · · b i +2 ,n · · · · · · · · · · · · · a n + ( − n ⊗ ε ( b ,n ) a n a b ,n b , · · · b i,n b ,i · · · b n − ,n b ,n − a · · · b ,i · · · b ,n − · · · · · · · · · · · · a i · · · b i,n − · · · · · · · · · · · · · · · a n − . The homology of the complex (cid:0) C • ( A, B, ε ) , ∂ (cid:1) is called the secondary Hochschild homology associ-ated to the triple ( A, B, ε ) and denoted by HH • ( A, B, ε ). The homology HH • ( A, B, ε ) is different
APURBA DAS, SATYENDRA KUMAR MISHRA, AND ANITA NAOLEKAR from the notion of secondary Hochschild homology of the triple (
A, B, ε ) with coefficients in themodule A . For the definition of HH • ( A, B, ε ), the secondary Hochschild cohomology associated toa triple (
A, B, ε ), one should see the detailed construction in Section 4 of [8]. Here, we skip thedefinition of HH • ( A, B, ε ) since we do not use it in the present article.4.
Calculus structure on secondary Hochschild (co)homology
In this section, we define comp module action of O • = C • (( A, B, ε ); A ) on the secondary Hochschildcomplex C • ( A, B, ε ). We prove that this comp module action makes C • ( A, B, ε ) a cyclic comp mod-ule over the operad C • (( A, B, ε ); A ). We conclude that (cid:0) H • (( A, B, ε ); A ) , HH • ( A, B, ε ) (cid:1) forms a(noncommutative) differential calculus.4.1. Cyclic comp module structure on C • ( A, B, ε ) . We denote M n := C n ( A, b, ε ) , for n ≥ k -linear maps • i : O m ⊗ M p → M p − m +1 as follows f • i ⊗ a b , · · · b ,p − b ,p a · · · b ,p − b ,p · · · · · · · · a p − b p − ,p · · · a p (9) = ⊗ a · · · b ,i − m + i − Q j = i b ,j b ,m + i · · · b ,p · · · b ,i − m + i − Q j = i b ,j b ,m + i · · · b ,p · · · · · · · · · · · · · · a i − m + i − Q j = i b i − ,j b i − ,m + i · · · b i − ,p · · · f (cid:0) T im + i − (cid:1) m + i − Q j = i b j,m + i · · · m + i − Q j = i b j,p · · · a m + i · · · b m + i,p · · · · · · · · · · · · · · · · · a p , for f ∈ O m , ≤ m ≤ p , and 0 ≤ i ≤ p − m + 1.Let f ∈ O m , g ∈ O n , and T ∈ M p . Then, for m ≥ , n, p ≥ , we first show that(10) f • i ( g • j T ) = g • j ( f • i + n − T ) if j < i, ( f ◦ j − i +1 g ) • i T if j − m < i ≤ j,g • j − m +1 ( f • i T ) if 0 ≤ i ≤ j − m. The left hand side is given by(11) f • i ( g • j T ) = f • i ⊗ a · · · b ,j − n + j − Q t = j b ,t b ,n + j · · · b ,p · · · b ,j − n + j − Q t = j b ,t b ,n + j · · · b ,p · · · · · · · · · · · · · · a j − n + j − Q t = j b j − ,t b j − ,n + j · · · b j − ,p · · · g (cid:0) T jn + j − (cid:1) n + j − Q s = j b s,n + j · · · n + j − Q s = j b s,p · · · a n + j · · · b n + j,p · · · · · · · · · · · · · · · · · a p . ALCULUS STRUCTURE ON SECONDARY HOCHSCHILD (CO)HOMOLOGY 9
Case 1 : For j < i , we have the following expression f • i ( g • j T )= f • i ⊗ a · · · b ,j − n + j − Q t = j b ,t · · · i + m + n − Q t = i + n − b ,t · · · b ,p · · · · · · · · · · · · · · · · · a j − n + j − Q t = j b j − ,t · · · i + m + n − Q t = i + n − b j − ,t · · · b j − ,p · · · g (cid:0) T jn + j − (cid:1) · · · n + j − Q s = j i + m + n − Q t = i + n − b s,t · · · n + j − Q s = j b s,p · · · · · · · · · · · · · · · · · · · · f (cid:0) T i + n − i + m + n − (cid:1) · · · i + m + n − Q s = i + n − b s,p · · · · · · · · · · · · · · · · · · · · · · · a p = g • j ⊗ a · · · b ,j − i + m + n − Q t = i + n − b ,t · · · b ,p · · · · · · · · · · · · · a i + n − i + m + n − Q t = i + n − b j − ,t · · · b j − ,p · · · f (cid:0) T i + n − i + m + n − (cid:1) · · · i + m + n − Q s = i + n − b s,p · · · · · · · · · · · · · · · · a p = g • j f • i + n − ⊗ a b , · · · b ,p − b ,p a · · · b ,p − b ,p · · · · · · · · a p − b n − ,p · · · a p ! = g • j ( f • i + n − T ) . By a similar calculation, the following cases hold.
Case 2:
For j − m < i ≤ j , we get f • i ( g • j T ) = ( f ◦ j − i +1 g ) • i T .
Case 3 : For 0 ≤ i ≤ j − m , we get f • i ( g • j T ) = g • j − m +1 ( f • i T ) . Unitality Condition:
For = Id A ∈ O = C (( A, B, ε ); A ), it is clear that • i T = T, for T ∈ M n , and 0 ≤ i ≤ n. We define a map t : M p → M p as follows(12) t ⊗ a b , · · · b ,p − b ,p a · · · b ,p − b ,p · · · · · · · · a p − b n − ,p · · · a p = ⊗ a p b ,p b ,p · · · b p − ,p a b , · · · b ,p − a · · · b ,p − · · · · · · · · a p − . Next, we show that the map t : M p → M p makes the unital comp module M • , a cyclic compmodule over the opeard O • . For T ∈ M p and f ∈ O • , we have the following expression t (cid:0) f • i T (cid:1) = t ⊗ a · · · b ,i − i + m − Q t = i b ,t · · · b ,p · · · · · · · · · · · · · a i − i + m − Q t = i b i − ,t · · · b i − ,p · · · f (cid:0) T ii + m − (cid:1) · · · m + i − Q s = i b s,p · · · · · · · · · · · · · · · · a p = ⊗ a p b ,p · · · b i − ,p m + i − Q s = i b s,p · · · b p − ,p a · · · b ,i − i + m − Q t = i b ,t · · · b ,p − · · · · · · · · · · · · · · a i − i + m − Q t = i b i − ,t · · · b i − ,p − · · · f (cid:0) T ii + m − (cid:1) · · · m + i − Q s = i b s,p − · · · · · · · · · · · · · · · · · a p − = f • i +1 ⊗ a p b ,p b ,p · · · b p − ,p a b , · · · b ,p − a · · · b ,p − · · · · · · · · a p − = f • i +1 t ( T ) . For i ≥ t i ( T ) = ⊗ a p − i +1 b p − i +1 ,p − i +2 · · · b p − i +1 ,p b ,p − i +1 b ,p − i +1 · · · b p − i,p − i +1 a p − i +2 · · · b p − i +2 ,p b ,p − i +2 b ,p − i +2 · · · b p − i,p − i +2 · · · · · · · · · · · a p b ,p b ,p · · · b p − i,p · · · a b , · · · b ,p − i · · · a · · · b ,p − i · · · · · · · · · · · · · · a p − i Clearly, it follows that the map t : M p → M p satisfies the identity t p +1 = Id . Thus, t p +1 = Id and t (cid:0) f • i T (cid:1) = f • i +1 t ( T ) . Therefore, by the above discussion we have the following theorem.4.1.
Theorem.
Let ( A, B, ε ) be a triple and O • := {O n } n ≥ be the operad structure on the complex C • (( A, B, ε ); A ) . Then the complex C • ( A, B, ε ) associated to the triple ( A, B, ε ) is a cyclic compmodule over the operad O • . (cid:3) Cyclic k -module structure on C • ( A, B, ε ) . The complex M • = C • ( A, B, ε ) is a cyclic compmodule over the multiplicative operad ( O • = C • (( A, B, ε ); A ) , µ ). It yields the following cyclic k -module structure on C • ( A, B, ε ):(i) for i = 0 , , . . . , p, the face maps d i : M p → M p − are given as follows ALCULUS STRUCTURE ON SECONDARY HOCHSCHILD (CO)HOMOLOGY 11 (a) if 0 ≤ i ≤ p − d i ( T ) = ⊗ a · · · b ,i − b ,i b ,i +1 · · · b ,p · · · · · · · · · a i − b i − ,i b i − ,i +1 · · · b i − ,p · · · a i a i +1 ε ( b i,i +1 ) · · · b i,n b i +1 ,n · · · · · · · · · · · · a p , (b) if i = p , d p ( T ) = ⊗ ε ( b ,n ) a n a b ,p b , · · · b p − ,p b ,p − a · · · b ,p − · · · · · · · b p − ,p − · · · a p − ;(ii) for j = 0 , , · · · , p, the degeneracies s j : M p → M p +1 are given by s j ⊗ a b , · · · b ,p a · · · b ,p · · · · · · · b p − ,p · · · a p = ⊗ a b , · · · b ,j · · · b ,p a · · · b ,j − · · · b ,p · · · · · · · · · · · a j · · · b j,p · · · A · · ·
11 1 · · · a j +1 · · · b j +1 ,p · · · · · · · · · · · · · · a p ;(iii) the cyclic operator t : M p → M p is given by the equation (12).The simplicial boundary operator on the complex M • is given by b := p X i =0 ( − i d i . Note that the complex ( M • , b ) is the same as the secondary Hochschild chain complex associatedto the triple ( A, B, ε ), defined in Section 3. Thus, the associated homology H • ( M ) = HH • ( A, B, ε ) . Next, Let us recall that the extra degeneracy map, the norm operator and the cyclic differentialare given by s − ( T ) := t s p ( T ) = 1 A • T, N := n X p =0 ( − ip t i , and B := ( Id − t ) s − N. The normalised complex N ( M ) • is the quotient of the complex M • by the subcomplex spanned by { A • j +1 − , j = 0 , , · · · , p } . The cyclic differential B induces the following map on the normalisedcomplex:(13) B ( T ) = s − N ( T ) = p X i =0 ( − ip A • t i ( T ) , for T ∈ N ( M ) p . Cosimplicial k -module structure on the complex C • (( A, B, ε ); A ) . Now, we consider thecosimplicial k -module structure on O • associated to the multiplicative operad ( O • , µ ). Then the face maps d i : O p → O p +1 and s j : O p → O p − are given as follows d i ( f ) = µ ◦ f if i = 0 ,f ◦ i µ if 1 ≤ i ≤ p,µ ◦ f if i = p + 1and s j ( f ) = f ◦ i +1 A , for j = 0 , , · · · , p − . In particular, s j ( f ) ⊗ a b , · · · b ,p − a · · · b ,p − · · · · · · · b p − ,p − · · · a p − = f ⊗ a · · · b ,j · · · b ,p − · · · b ,j − · · · b ,p − · · · · · · · · · · a j − · · · b j − ,p − · · · A · · · · · · a j · · · b j,p − · · · · · · · · · · · · · a p − . The coboundary map δ p : O p → O p +1 is given by δ p := P p +1 i =0 d i . The cochain complex ( O • , δ )coincides with the complex ( C • (( A, B, ε ); A ) , δ ε ). The conormalised cochain complex is given by N ( O ) p := ∩ p − j =0 Ker ( s j ) . Here, N ( O ) p is the collection of maps in Hom k ( A ⊗ p ⊗ B ⊗ p ( p − , A ), which vanish at the elementsof the type ⊗ a · · · b ,j · · · b ,p − · · · b ,j − · · · b ,p − · · · · · · · · · · a j − · · · b j − ,p − · · · A · · · · · · a j · · · b j,p − · · · · · · · · · · · · · a p − . Note that the cohomology H • ( N ( O ) • ) = H • µ ( O • ) = H • (( A, B, ε ); A ).4.4. A calculus structure on (cid:0) H • (( A, b, ε ); A ) , HH • ( A, b, ε ) (cid:1) . The complex M • := C • ( A, B, ε )is a cyclic comp module over the multiplicative operad O = C • (( A, B, ε ); A ). Let us recall fromSection 2 that a cyclic (unital) comp module M • over a multiplicative operad ( O , µ ) induces acalculus structure on the pair (cid:0) H • µ ( O ) , H • ( M ) (cid:1) . From Subsections 4.2 and 4.3, it follows that H • ( M ) = HH • ( A, B, ε ) ,H • µ ( O • ) = H • (( A, B, ε ); A ) . Therefore, we obtain the following result.4.2.
Theorem.
The pair (cid:0) H • (( A, b, ε ); A ) , HH • ( A, b, ε ) (cid:1) forms a noncommutative differential cal-culus. (cid:3) ALCULUS STRUCTURE ON SECONDARY HOCHSCHILD (CO)HOMOLOGY 13 conclusion M. Staic first introduced secondary Hochschild (co)homology [12,13] in order to study the deforma-tion theory of associative algebras over a commutative ring. The secondary Hochschild (co)homologybehaves similar to the Hochschild (co)homology in several aspects. However, unlike Hochschild(co)homology, there is no natural cyclic action on the (co)chain complex. The absence of a naturalcyclic action also explains the natural constructions of new complexes while introducing secondarycyclic (co)homology [8].In the Hochschild case, there is a noncommutative differential calculus ( HH • ( A, A ) , HH • ( A, A )).Under certain conditions [5], the isomorphism HH • ( A, A ) ∼ = HH • ( A, A ) yields a BV algebra struc-ture on the cohomology HH • ( A, A ). In fact, the condition obtained by [5] (in the case of Calabi-Yaualgebras) can be written in general for any calculus: if ( A , Ω) is a calculus, then i [ f,g ] = [ i f , L g ]= i f ◦ L g − ( − m ( n +1) L g ◦ i f = i f ◦ ( B ◦ i g − ( − n i g ◦ B ) − ( − m ( n +1) ( B ◦ i g − ( − n i g ◦ B ) ◦ i f = i f ◦ B ◦ i g − ( − n i f⌣g ◦ B − ( − m ( n +1) ( − mn B ◦ i f⌣g + ( − m ( n +1)+ n i g ◦ B ◦ i f (14)for f ∈ A m , g ∈ A n . Subsequently, we obtain the following result which gives a condition on thecalculus such that the underlying Gerstenhaber algebra is a Batalin-Vilkovisky (BV) algebra.5.1. Theorem.
Let ( A , Ω) be a calculus. If there exists an element c ∈ Ω k such that B ( c ) = 0 andthe map Θ : A • → Ω k −• , given by Θ( f ) = i f c is an isomorphism, then the map ∆ : A • → A •− , defined by i ∆ f c = B ( i f c ) makes the Gerstenhaber algebra A into a BV algebra.Proof. Since B ( c ) = 0 and i ∆ f c = B ( i f c ), by equation (14) it follows that i [ f,g ] ( c ) = − ( − m (cid:0) i ∆( f⌣g ) c − i ∆( f ) ⌣g c − ( − m i f⌣ ∆( g ) c (cid:1) . In turn the condition that Θ is an isomorphism implies that the operator ∆ generates the Gersten-haber bracket on A , i.e.[ f, g ] = − ( − m (cid:0) ∆( f ⌣ g ) − ∆( f ) ⌣ g − ( − m f ⌣ ∆( g ) (cid:1) , for all f ∈ A m , g ∈ A n . (cid:3) Thus, Theorem 5.1 gives a condition on the calculus (cid:0) H • (( A, b, ε ); A ) , HH • ( A, b, ε ) (cid:1) such that theGerstenhaber algebra structure on H • (( A, b, ε ) , A ) becomes a BV algebra structure. More precisely,if we have an element [ T ] ∈ HH • ( A, B, ε ) such that B [ T ] = 0 and the mapΘ : H • (( A, b, ε ) , A ) → HH k −• ( A, B, ε ) , given by Θ( f ) = i f [ T ]is an isomorphism, then H • (( A, b, ε ) , A ) is a BV algebra.In the Hochschild case, if A is symmetric algebra then the calculus ( HH • ( A, A ) , HH • ( A, A ))satisfies the conditions in the Theorem 5.1 and the Hochschild cohomology HH • ( A, A ) carries aBV algebra structure (see [9,10,16] for details). It will be interesting to find conditions on the B -algebra A such that the conditions in Theorem 5.1 hold. It needs further investigation since 1)there is no derived functor description for the secondary Hochschild (co)homology, and 2)- even inthe finite-dimensional symmetric algebra case, the most canonical choice for the BV-operator doesnot lift to the secondary Hochschild cohomology [1]. More precisely, in [1] the authors consider afinite-dimensional symmetric algebra A and define a BV-operator ∆ on the homotopy Gerstenhaberalgebra C ∗ (( A, B, ε ) , A ), which due to proposition 9 and theorem 10 in [1], is the most canonicalchoice for a square zero BV-differential operator on H ∗ (( A, B, ε ) , A ). Though ∆ determines thegraded Lie bracket on H ∗ (( A, B, ε ) , A ), it is not a cochain map in general (see theorem 7, [1]). References [1] M. Balodi, A. Banerjee, A. Naolekar, “BV-operators and the secondary Hochschild complex”,
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Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur 208016, UttarPradesh, India.
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