Lie brackets on Hopf algebra cohomology
aa r X i v : . [ m a t h . R A ] J a n LIE BRACKETS ON HOPF ALGEBRA COHOMOLOGY
TEK˙IN KARADA ˘G AND SARAH WITHERSPOON
Abstract.
By work of Farinati, Solotar, and Taillefer, it is known that the Hopf alge-bra cohomology of a quasi-triangular Hopf algebra, as a graded Lie algebra under theGerstenhaber bracket, is abelian. Motivated by the question of whether this holds fornonquasi-triangular Hopf algebras, we show that Gerstenhaber brackets on Hopf algebracohomology can be expressed via an arbitrary projective resolution using Volkov’s ho-motopy liftings as generalized to some exact monoidal categories. This is a special caseof our more general result that a bracket operation on cohomology is preserved underexact monoidal functors—one such functor is an embedding of Hopf algebra cohomologyinto Hochschild cohomology. As a consequence, we show that this Lie structure on Hopfalgebra cohomology is abelian in positive degrees for all quantum elementary abeliangroups, most of which are nonquasi-triangular. Introduction
The Ext algebra of a Hopf algebra, that is its Hopf algebra cohomology, has a Liestructure arising from its embedding into Hochschild cohomology in case the antipodeis bijective. This Lie structure is known to be abelian when the Hopf algebra is quasi-triangular. More specifically, Taillefer [10] constructed a Lie bracket on cohomology arisingfrom a category of Hopf bimodule extensions of a Hopf algebra and showed that brackets arealways zero. In the finite dimensional case, this corresponds to Hopf algebra cohomology ofthe opposite of the Drinfeld double, whose Lie structure was also investigated by Farinatiand Solotar [2] using different techniques. Farinati and Solotar showed this is indeed thebracket arising from an embedding into Hochschild cohomology. Hermann [4] looked ata more general monoidal category setting and showed that the Lie structure is trivial incase the category is braided [4, Theorem 5.2.7]. It is an open question as to what happensmore generally.In this paper, we apply a new technique, the homotopy lifting method of Volkov [11],to shed more light on this question, particularly in the Hopf algebra setting. Homotopyliftings were defined for some exact monoidal categories in [12], and we use them to provethat brackets are preserved under exact monoidal functors. As a consequence, we showthat the Lie bracket on Hopf algebra cohomology defined by homotopy liftings agreeswith that induced by its embedding into Hochschild cohomology, in case the antipode isbijective. An advantage of the homotopy lifting method is its ability to handle the Liestructure independently of choice of projective resolution. A good choice of resolution canfacilitate understanding of the Lie structure.
Date : January 23, 2021.
Key words and phrases.
Hochschild cohomology, Hopf algebra cohomology, Gerstenhaber brackets, ho-motopy lifting.Partially supported by NSF grants 1665286 and 2001163.
We illustrate with the quantum elementary abelian groups. These are tensor productsof copies of the Taft algebras T n that depend on an n th root of unity; when n > T n is trivial by using techniques of Negron and the secondauthor [7], a consequence of a larger computation of the bracket on Hochschild cohomologyof T n . This was the first known example of the Lie structure on Hopf algebra cohomologyof a nonquasi-triangular Hopf algebra. In this paper we take a different approach, givinga second proof in positive degrees, for Taft algebras T n via homotopy liftings instead ofrelying on a computation for the full Hochschild cohomology ring. We extend this resultto quantum elementary abelian groups, as iterated tensor products of Taft algebras.An outline of the contents is as follows. In Section 2, we recall some definitions andthe homotopy lifting method for defining a Lie structure on the cohomology of some exactmonoidal categories. We use this method in Section 3, proving that the Lie structure isalways preserved under an exact monoidal functor. In Section 4, we apply the results ofSection 3 to Hopf algebra cohomology, concluding a formula for Gerstenhaber bracketsbased on homotopy liftings. The resulting Lie structure on the Hopf algebra cohomologyof a quantum elementary abelian group, in positive degrees, is shown to be abelian inSection 5. 2. Preliminaries
We begin by recalling the definitions of Hopf algebra cohomology and Hochschild coho-mology, as well as an embedding of one into the other that provides a connection betweentheir Lie structures.Let k be a field and abbreviate ⊗ k by ⊗ when it will cause no confusion. Hopf algebra cohomology.
Let A be a Hopf algebra over a field k with coproduct∆ : A → A ⊗ A , counit ε : A → k and antipode S : A → A . We will always assume that S is bijective. Note that, since all finite dimensional and most known infinite dimensionalHopf algebras of interest have bijective antipodes, this assumption is not very restrictive.Give the field k the structure of an A -module, called the trivial A -module, via the counit ε , i.e. a · k = ε ( a ) for all a ∈ A . Give A the structure of an A -bimodule by multiplicationon both sides; equivalently, A is a left A e -module where A e := A ⊗ A op and A op is thevector space A with multiplication opposite to A . The Hopf algebra cohomology of A isH ∗ ( A, k ) := Ext ∗ A ( k, k ). The Hochschild cohomology of A is HH ∗ ( A ) := Ext ∗ A e ( A, A ).We can embed Hopf algebra cohomology into Hochschild cohomology. We will needdetails, and we recall some lemmas. For proofs, see [13, Section 9.4].
Lemma 2.1.
There is an isomorphism of A e -modules, A ∼ = A e ⊗ A k, where A e ⊗ A k is the tensor induced A e -module under the identification of A with thesubalgebra of A e that is the image of the embedding δ : A → A e defined for all a ∈ A by δ ( a ) = X a ⊗ S ( a ) . Lemma 2.2.
The right A -module A e , where A acts by right multiplication under its iden-tification with δ ( A ) , is projective. OPF ALGEBRA COHOMOLOGY 3
Lemma 2.3 (Eckmann-Shapiro) . Let A be a ring and let B be a subring of A such thatA is projective as a right B -module. Let M be an A -module and N be a B -module. Then Ext nB ( N, M ) ∼ = Ext nA ( A ⊗ B N, M ) , where M is considered to be a B -module under restriction. We will consider A to be a left A -module by the left adjoint action, which is for a, b ∈ A , a · b = X a bS ( a ) . Denote this A -module by A ad .For any left A -module M , let H ∗ ( A, M ) := Ext ∗ A ( k, M ). The following theorem is wellknown; see, e.g. [13, Theorem 9.4.5]. We sketch a proof since we will need some of thedetails later. Theorem 2.4.
There is an isomorphism of graded k -vector spaces HH ∗ ( A ) ∼ = H ∗ ( A, A ad ) . Proof.
By Lemma 2.2, A e is a projective as a right A -module, so we can apply theEckmann-Shapiro Lemma. We replace A with A e , B with A and take M = A , N = k in the Eckmann-Shapiro Lemma and obtain the isomorphism Ext nA e ( A e ⊗ A k, A ) ∼ =Ext nA ( k, A ad ) as k -vector spaces. Lastly, we apply Lemma 2.1 and obtain Ext nA e ( A, A ) ∼ =Ext nA ( k, A ad ). (cid:3) A consequence of the theorem is an embedding of Hopf algebra cohomology H ∗ ( A, k ) intoHochschild cohomology HH ∗ ( A ): Let P → k be a projective resolution of the A -module k . We can embed H ∗ ( A, k ) into H ∗ ( A, A ad ) ∼ = HH ∗ ( A ) via the map η ∗ : Hom A ( P, k ) → Hom A ( P, A ad ) induced by the unit map η : k → A (see [13, Corollary 9.4.7]). Equivalently,viewing each space H n ( A, k ) as equivalence classes of n -extensions of k by k , the functor A e ⊗ A − induces an embedding of H ∗ ( A, k ) into HH ∗ ( A ). Homotopy liftings and exact monoidal categories.
Gerstenhaber defined the bracketon Hochschild cohomology via the bar resolution. We will not need that definition here.Instead, by using homotopy liftings as defined below, a similar operation is constructedfor some exact monoidal categories in [12, Section 4] as follows. This turns out to beequivalent to the Gerstenhaber bracket when the category is that of A -bimodules.We refer to [1] for definitions and properties of monoidal categories. Let C be an exactmonoidal category and let be its unit object. As is customary, we will identify ⊗ X and X ⊗ with X for all objects X in C , under assumed fixed isomorphisms (for whichwe will not need notation). One example of an exact monoidal category is the category ofleft modules for a Hopf algebra over k , with tensor product ⊗ of modules, and the unitobject given by the trivial module k . Another example is the category of A -bimodules(equivalently left A e -modules) for an associative algebra A over k , with tensor product ⊗ A and unit object A .Let P → be a projective resolution of with differential d and let µ P : P → be thecorresponding augmentation map. In [12, Definition 4.3], the resolution ( P, d, µ P ) of iscalled n - power flat if ( P ⊗ r , d ⊗ r , µ ⊗ rP ) is a projective resolution of for each r (1 ≤ r ≤ n ).If P is n -power flat for each n ≥
2, then we say that P is power flat . For the two categoriesmentioned above, projective resolutions of are generally power flat. TEK˙IN KARADA ˘G AND SARAH WITHERSPOON
Assume has a projective power flat resolution P . For a degree l morphism ψ : P → P ,its differential in the Hom complex Hom C ( P, P ) is defined to be ∂ ( ψ ) = dψ − ( − l ψd. Let f : P → be an m -cocycle. Let ∆ P : P → P ⊗ P be a diagonal map, i.e. a chainmap lifting the isomorphism ∼ −→ ⊗ . A degree ( m −
1) morphism ψ f : P → P is a homotopy lifting of ( f, ∆ P ) if(2.5) ∂ ( ψ f ) = ( f ⊗ P − P ⊗ f )∆ P and µ P ψ f ∼ ( − m +1 f ψ for some degree − ψ : P → P such that(2.6) ∂ ( ψ ) = ( µ P ⊗ P − P ⊗ µ P )∆ P . The cohomology of the monoidal category C is H ∗ ( C ) = H ∗ ( C , ) := Ext ∗C ( , ). It hasa Lie bracket defined as follows [12, Section 4]: For an m -cocycle f : P m → and an n -cocycle g : P n → , let ψ f and ψ g be homotopy liftings of ( f, ∆ P ) and ( g, ∆ P ) respectively.Then the cochain [ f, g ] defined as(2.7) [ f, g ] = f ψ g − ( − ( m − n − gψ f induces a graded Lie bracket on H ∗ ( C ). That is, it induces a well-defined operation oncohomology that is graded alternating and satisfies a graded Jacobi identity (cf. [13, Lemma1.4.3]). 3. Change of exact monoidal categories
Let C , C ′ be exact monoidal categories for which there exist power flat resolutions oftheir unit objects , ′ . Let F : C → C ′ be an exact monoidal functor [1, Definition 2.4.1],that is, F is exact and there is a natural isomorphism η of functors from C × C to C ′ givenby η X,Y : F ( X ) ⊗ F ( Y ) → F ( X ⊗ Y )for all X , Y in C , F ( ) ∼ = ′ and ( F, η ) satisfies the monoidal structure axiom of [1,Definition 2.4.1], that is, the following diagram commutes for all objects
X, Y, Z in C :(3.1) ( F ( X ) ⊗ F ( Y )) ⊗ F ( Z ) η X,Y ⊗ F ( Z ) (cid:15) (cid:15) ∼ / / F ( X ) ⊗ ( F ( Y ) ⊗ F ( Z )) F ( X ) ⊗ η Y,Z (cid:15) (cid:15) F ( X ⊗ Y ) ⊗ F ( Z ) η X ⊗ Y,Z (cid:15) (cid:15) F ( X ) ⊗ F ( Y ⊗ Z ) η X,Y ⊗ Z (cid:15) (cid:15) F (( X ⊗ Y ) ⊗ Z ) ∼ / / F ( X ⊗ ( Y ⊗ Z ))(The horizontal isomorphisms are given by the associativity constraint for C ′ and the imageof the associativity constraint for C under F , respectively. We will not need notation forthese isomorphisms.)Denote the isomorphism from F ( ) to ′ by φ . Then the following diagrams commutefor all objects X by [1, Proposition 2.4.3]; we have chosen to show diagrams involving theinverse maps η − ,X and η − X, since we will need these later. The unlabeled isomorphisms OPF ALGEBRA COHOMOLOGY 5 in the diagrams are those canonically determined by the fixed isomorphisms given bytensoring with unit objects and the fixed isomorphism F ( ) ∼ = ′ . F ( ⊗ X ) ∼ / / η − ,X (cid:15) (cid:15) F ( X ) ∼ (cid:15) (cid:15) F ( X ⊗ ) ∼ / / η − X, (cid:15) (cid:15) F ( X ) ∼ (cid:15) (cid:15) F ( ) ⊗ F ( X ) ∼ / / ′ ⊗ F ( X ) F ( X ) ⊗ F ( ) ∼ / / F ( X ) ⊗ ′ Example 3.2.
Let A be a Hopf algebra and B a Hopf subalgebra of A . Let C be thecategory of left A -modules and C ′ the category of left B -modules. Let F : C → C ′ bethe restriction functor, that is on each A -module X , the action is restricted to B via theinclusion map B ֒ → A . The restriction of a tensor product of modules to B is isomorphicto the tensor product of their restrictions to B , and thus there is a natural transformation η as required.In the next section we will apply the following theorem to shed light on the connectionbetween the Lie structures on Hopf algebra cohomology and on Hochschild cohomology.Let P be a projective resolution of in C and write P ′ = F ( P ), which is a projectiveresolution of F ( ) ∼ = ′ in C ′ under our assumptions. Let d denote the differential and µ P : P → denote the augmentation map of P . Write d ′ = F ( d ) and µ P ′ = F ( µ P ).Note that P ⊗ P is also a projective resolution of in C with augmentation map µ P ⊗ µ P followed by the canonical isomorphism ⊗ ∼ −→ . Let∆ P ′ = η − P,P F (∆ P ) , which is a diagonal map on P ′ under our assumptions. Theorem 3.3.
Let C , C ′ be exact monoidal categories and let F : C → C ′ be an exactmonoidal functor. Assume there exists a power flat resolution P of in C . Let f ∈ Hom C ( P m , ) , an m -cocycle. Let ψ f be a homotopy lifting of f with respect to ∆ P . Then F ( ψ f ) is a homotopy lifting of F ( f ) with respect to ∆ P ′ .Proof. Since ψ f is a homotopy lifting of f , dψ f − ( − m − ψ f d = ( f ⊗ − ⊗ f )∆ P . Set f ′ = F ( f ), ψ f ′ = F ( ψ f ), and apply F to each side of this equation to obtain(3.4) d ′ ψ f ′ − ( − m − ψ f ′ d ′ = F ( f ⊗ − ⊗ f ) F (∆ P ) . Since η is a natural transformation, under our assumptions (see the above commutingdiagrams), the following diagram commutes: F ( P ) F (∆ P ) / / F ( P ⊗ P ) F ( f ⊗ / / η − P,P (cid:15) (cid:15) F ( ⊗ P ) ∼ / / η − ,P (cid:15) (cid:15) F ( P ) = (cid:15) (cid:15) F ( P ) ⊗ F ( P ) F ( f ) ⊗ / / F ( ) ⊗ F ( P ) ∼ / / F ( P )Therefore F ( f ⊗ F (∆ P ) can be identified with ( F ( f ) ⊗ η − P,P F (∆ P ), and similarly F (1 ⊗ f ) F (∆ P ) with (1 ⊗ F ( f )) η − P,P F (∆ P ). So the right side of expression (3.4) is equal TEK˙IN KARADA ˘G AND SARAH WITHERSPOON to ( f ′ ⊗ − ⊗ f ′ ) η − P,P F (∆ P ) , which is in turn equal to ( f ′ ⊗ − ⊗ f ′ )∆ P ′ , as desired.Since ψ f is a homotopy lifting, µ P ψ f ∼ ( − m +1 f ψ for some degree − ψ : P → P such that ∂ ( ψ ) = dψ + ψd = ( µ P ⊗ P − P ⊗ µ P )∆ P . By applying F to both sides of the above equation, we obtain F ( dψ + ψd ) = F ( µ P ⊗ P − P ⊗ µ P ) F (∆ P ) , and under our identifications, letting ψ ′ = F ( ψ ), this is d ′ ψ ′ + ψ ′ d ′ = F ( µ P ⊗ F (∆ P ) − F (1 ⊗ µ P ) F (∆ P ) . Via a commutative diagram such as that above, we see this is equal to( µ P ′ ⊗ P ′ − (1 ⊗ µ P ′ )∆ P ′ . Therefore, ψ ′ : P ′ → P ′ is a degree − ∂ ( ψ ′ ) = d ′ ψ ′ + ψ ′ d ′ = ( µ P ′ ⊗ P ′ − (1 ⊗ µ P ′ )∆ P ′ , and µ P ′ F ( ψ f ) ∼ ( − m +1 F ( f ) ψ ′ , that is, µ P ′ ψ f ′ ∼ ( − m +1 f ′ ψ ′ , as desired.We have shown that F ( ψ f ) is a homotopy lifting of F ( f ) with respect to ∆ P ′ . (cid:3) Corollary 3.5.
The functor F induces a graded Lie algebra homomorphism from H ∗ ( C ) to H ∗ ( C ′ ) , in positive degrees.Proof. As a consequence of the theorem and formula (2.7), the functor F takes the Liebracket of two elements of positive degree in the cohomology H ∗ ( C ) to the Lie bracket oftheir images in H ∗ ( C ′ ) under F . (cid:3) Hopf algebra cohomology
Let A be a Hopf algebra with bijective antipode. Let C be the category of (left) A -modules, and let C ′ be the category of (left) A e -modules. For each A -module U , let F ( U ) = A e ⊗ A U, the tensor induced module, where we identify A with the subalgebra δ ( A ) of A e as inLemma 2.1. Also by Lemma 2.1, F takes the unit object k of C to an isomorphic copy ofthe unit object A of C ′ . It takes projective A -modules to projective A e -modules since A e isprojective as an A -module by Lemma 2.2. For each A -module homomorphism f : U → V ,define F ( f ) by F ( f )((1 ⊗ ⊗ A u ) = (1 ⊗ ⊗ A f ( u )for all u ∈ U . Then F may be viewed as the functor providing the embedding of Hopfalgebra cohomology H ∗ ( A, k ) into Hochschild cohomology HH ∗ ( A ); see the proof of Theo-rem 2.4 and the subsequent paragraph.For each pair of A -modules U , V , we wish to define an A e -module homomorphism η U,V : F ( U ) ⊗ A F ( V ) → F ( U ⊗ V ) , that is, η U,V : ( A e ⊗ A U ) ⊗ A ( A e ⊗ A V ) → A e ⊗ A ( U ⊗ V ) . OPF ALGEBRA COHOMOLOGY 7
For all a, b ∈ A , u ∈ U , and v ∈ V , set η U,V (( a ⊗ ⊗ A u ) ⊗ A ((1 ⊗ b ) ⊗ A v ) = ( a ⊗ b ) ⊗ A ( u ⊗ v ) . Note that all elements of ( A e ⊗ A U ) ⊗ A ( A e ⊗ A V ) can indeed be written as linear combi-nations of elements of the indicated forms and that the map is well-defined. For example,for all a, b ∈ A and u ∈ U , letting b = S ( b ′ ),( a ⊗ b ) ⊗ A u = X ( aS ( b ′ ) b ′ ⊗ S ( b ′ )) ⊗ A u = X ( aS ( b ′ ) ⊗ ⊗ A (( b ′ ⊗ S ( b ′ )) · u ) . By its definition, η U,V is an A e -module homomorphism.We check that η is a natural transformation. That is, the following diagram commutesfor all objects U, V, U ′ , V ′ and morphisms f : U → U ′ , g : V → V ′ : F ( U ) ⊗ A F ( V ) η U,V (cid:15) (cid:15) F ( f ) ⊗ A F ( g ) / / F ( U ′ ) ⊗ A F ( V ′ ) η U ′ ,V ′ (cid:15) (cid:15) F ( U ⊗ V ) F ( f ⊗ g ) / / F ( U ′ ⊗ V ′ )Commutativity follows from the definitions of η U,V , η U ′ ,V ′ . To see that η is monoidal, thatis diagram (3.1) commutes, it is easier to check the corresponding diagram associated to η − , a straightforward calculation.For the following theorem, we define the Gerstenhaber bracket of two elements in Hopfalgebra cohomology H ∗ ( A, k ) via the embedding into Hochschild cohomology followed bythe Gerstenhaber bracket on Hochschild cohomology. The theorem states that this is thesame as their bracket defined by (2.7) on Hopf algebra cohomology H ∗ ( A, k ) via homo-topy liftings. Thus the theorem allows us to bypass the need to work with Hochschildcohomology at all, for questions purely about Hopf algebra cohomology.
Theorem 4.1.
Let A be a Hopf algebra with bijective antipode. Let P be a projectiveresolution of k and let f , g be cocycles in Hom A ( P m , k ) , Hom A ( P n , k ) , respectively, repre-senting elements of Hopf algebra cohomology H ∗ ( A, k ) . Let ∆ P be a diagonal map, and let ψ f , ψ g be homotopy liftings of f, g with respect to ∆ P . The Gerstenhaber bracket of thecorresponding elements in Hopf algebra cohomology H ∗ ( A, k ) is represented by [ f, g ] = f ψ g − ( − ( m − n − gψ f . Proof.
This is an immediate consequence of Theorem 3.3 and expression (2.7), since weshowed above that the induction functor F is an exact monoidal functor. (cid:3) One consequence Theorem 4.1 is a quick new proof that for a cocommutative Hopfalgebra in characteristic not 2, Gerstenhaber brackets on Hopf algebra cohomology inpositive degree are always 0. We state this as Corollary 4.2 next. Since cocommutativeHopf algebras are quasi-triangular, this is a small special case of the well known resultsof Farinati, Solotar, Taillefer, and Hermann, but it highlights our completely differentapproach.
Corollary 4.2.
Let k be a field of characteristic not 2, and let A be a cocommutative Hopfalgebra. The Lie structure on Hopf algebra cohomology H ∗ ( A, k ) , given by the Gerstenhaberbracket, is abelian in positive degrees. TEK˙IN KARADA ˘G AND SARAH WITHERSPOON
Proof.
Let P be a projective resolution of k as an A -module. Let ∆ ′ : P → P ⊗ P be adiagonal map. Let σ : P ⊗ P → P ⊗ P be the signed transposition map, i.e. σ ( x ⊗ y ) =( − | x || y | y ⊗ x for all homogeneous x, y ∈ P . Since A is cocommutative, σ ∆ ′ is also an A -module homomorphism, and therefore a diagonal map. Let∆ = 12 (∆ ′ + σ ∆ ′ ) , a diagonal map as well. Note that ∆ is symmetric in the sense that σ ∆ = ∆.Now, by symmetry, ( µ P ⊗ P − P ⊗ µ P )∆ ≡
0, and so in (2.6), we can take ψ ≡
0. Similarly, in (2.5), we can take ψ f ≡ f . Thus by Theorem 4.1,Gerstenhaber brackets on the Hopf algebra cohomology H ∗ ( A, k ) are always 0 in positivedegrees. (cid:3) Taft algebras and quantum elementary abelian groups
In this section we illustrate our results by showing that the Lie structure on the Hopfalgebra cohomology of a Taft algebra, and more generally of a quantum elementary abeliangroup, is abelian in positive degrees.Let k be a field of characteristic 0 containing a primitive n th root ω of 1. Let A be theTaft algebra generated by x and g with relations gx = ωxg, x n = 0 , g n = 1 . We take ∆( x ) = x ⊗ g ⊗ x, ∆( g ) = g ⊗ g,ε ( x ) = 0, ε ( g ) = 1, S ( x ) = − g − x , S ( g ) = g − . Let B = k [ x ] / ( x n ), a subalgebra of A , andlet P be the following projective resolution of the trivial B -module k : · · · x n − · / / B x · / / B x n − · / / B x · / / B ε / / k / / B on each term is by multiplication. Give each component B in the resolutionthe structure of an A -module by letting g · x i = ω i x i in even degrees and g · x i = ω i +1 x i in odd degrees. For clarity of notation, in each degree l , denote the element 1 B of P l = B by ǫ l . Note that P l is projective as an A -module since the characteristic of k is notdivisible by n : Specifically, in even degrees, there are A -module homomorphisms P l → A ( x i n P n − j =0 x i g j ) and A → P l ( x i g j x i ) whose composition is the identity map. Inodd degrees, a similar statement is true of the maps P l → A and A → P l given respectivelyby x i ( n P n − j =0 x i +1 g j , if i < n − , n P n − j =0 g j , if i = n − x i g j (cid:26) x i − , if i = 0 ,x n − , if i = 0 . Calculations show that the following formulas yield a diagonal map ∆ : P → P ⊗ P ,that is for each l , ∆ l is an A -module homomorphism, and ∆ is a chain map lifting the OPF ALGEBRA COHOMOLOGY 9 canonical isomorphism k ∼ −→ k ⊗ k :∆ j +1 ( ǫ j +1 ) = j +1 X i =0 ǫ i ⊗ ǫ j +1 − i , ∆ j ( ǫ j ) = j X i =0 ǫ i ⊗ ǫ j − i + j − X i =0 n − X a =0 (cid:18) n − a + 1 (cid:19) ω x a ǫ i +1 ⊗ x n − − a ǫ j − i − , where (cid:0) n − a +1 (cid:1) ω is the ω -binomial coefficient defined for all nonnegative integers a, b, c by (cid:18) bc (cid:19) ω = ( b ) ω ( b − ω · · · ( b − c + 1) ω ( c ) ω ( c − ω · · · (1) ω where ( a ) ω = 1 + ω + ω + · · · + ω a − . Note that ∆ = σ ∆ since (cid:0) n − a +1 (cid:1) ω = (cid:0) n − a (cid:1) ω in general. However, symmetry does holdafter projection onto even degrees, a key property for the proof of the theorem below sincethe cohomology is concentrated in even degrees as we see next.The cohomology of A can be computed directly from the resolution P above, and isH ∗ ( A, k ) ∼ = H ∗ ( B, k ) G ∼ = k [ z ] , where deg( z ) = 2. Alternatively, see [9, Corollary 3.4] for the relevant general theory forskew group algebras.The following theorem was proven in [5] by different techniques, and more generallythere, the elements of degree 0 were included. The homotopy lifting method that we usehere was designed for positive degree cohomology. Theorem 5.1.
The Lie structure given by the Gerstenhaber bracket on the cohomology H ∗ ( A, k ) of a Taft algebra A is abelian in positive degrees.Proof. Let P be the resolution of k given above. Let f ∈ Hom A ( P , k ) denote the cocyclewith f ( ǫ ) = 1, a representative of the generator z of the cohomology ring H ∗ ( A, k ),described above. By [13, Lemma 1.4.7], it will suffice to show that the bracket of f withitself is 0 since f represents an algebra generator of cohomology.We wish to find a homotopy lifting of f . First note that in (2.6), we can take ψ ≡ P ⊗ P i ) ⊕ ( P i ⊗ P ) for each i . Similarly, by symmetry of the image of the diagonal mapunder the projection onto ( P even ⊗ P ) ⊕ ( P ⊗ P even ), since f has even degree, in (2.5), wecan take ψ f ≡ ψ f must satisfy dψ f + ψ f d = ( f ⊗ − ⊗ f )∆ P . The right side of this equation, evaluated on ǫ l , is( f ⊗ − ⊗ f )(∆ P ( ǫ l )) , and comparing to the formulas for ∆ P ( ǫ j ) and ∆ P ( ǫ j +1 ) above, the only terms that willbe nonzero after applying f ⊗ − ⊗ f are those having ǫ as one of the tensor factors. Bysymmetry, the resulting terms after applying f ⊗ ⊗ f cancel due to their oppositesigns. So we may take ψ f ≡ f, f ] = 2 f ψ f = 0. (cid:3) The following theorem is a consequence since the Lie structure of a tensor product ofalgebras reduces to that on each factor [6]. Quantum elementary abelian groups are definedto be iterated tensor products of Taft algebras [8].
Theorem 5.2.
Let A be a quantum elementary abelian group. The Lie structure of theHopf algebra cohomology H ∗ ( A, k ) , given by Gerstenhaber bracket, is abelian in positivedegrees. References [1]
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Department of Mathematics, Texas A&M University, College Station, Texas 77843, USA
Email address : [email protected] Department of Mathematics, Texas A&M University, College Station, Texas 77843, USA
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