Gerstenhaber bracket on Hopf algebra and Hochschild cohomologies
GGerstenhaber bracket on Hopf algebra and Hochschildcohomologies
Tekin Karada˘gOctober 16, 2020
Abstract
We calculate the Gerstenhaber bracket on Hopf algebra and Hochschild coho-mologies of the Taft algebra T p for any integer p > T p , as in all known quasi-triangular Hopf algebras. This example is thefirst known bracket computation for a nonquasi-triangular algebra. Also, we finda general formula for the bracket on Hopf algebra cohomology of any Hopf algebrawith bijective antipode on the bar resolution that is reminiscent of Gerstenhaber’soriginal formula for Hochschild cohomology. Gerstenhaber brackets were originally defined on Hochschild cohomology by M. Ger-stenhaber himself [3, Section 1.1]. In 2002, A. Farinati and A. Solotar showed that forany Hopf algebra A , Hopf algebra cohomology H ∗ ( A ) := Ext ∗ A ( k, k ) is a Gerstenhaberalgebra [2]. Hence, we can define a Gerstenhaber bracket on Hopf algebra cohomology.In the same year, R. Taillefer used a different approach and found a bracket on Hopfalgebra cohomology [11] which is equivalent to the bracket constructed by A. Farinatiand A. Solotar. The category of A -modules and the category of A e -modules are exam-ples of strong exact monoidal categories. In 2016, Reiner Hermann [5, Theorem 6.3.12,Corollary 6.3.15] proved that if the strong exact monoidal category is lax braided, thenthe bracket is constantly zero. Therefore, the Gerstenhaber bracket on the Hopf algebracohomology of a quasi-triangular Hopf algebra is trivial. However, we do not know thebracket structure for a nonquasi-triangular Hopf algebra. Taft algebras are nice exam-ples of nonquasi-triangular Hopf algebras. In this paper, we show that the Gerstenhaberbracket on the Hochschild cohomology of a Taft Algebra is nontrivial. However, the Key words and phrases:
Hochschild cohomology, Hopf algebra cohomology, Gerstenhaber bracket,Taft algebraPartially supported by NSF grant 1665286. a r X i v : . [ m a t h . QA ] O c t racket structure on Hopf algebra cohomology of a Taft algebra is constantly zero. Also,we take the Gerstenhaber bracket formula on Hochschild comology and find a generalformula for Gerstenhaber bracket on Hopf algebra cohomology.We start by giving some basic definitions and some tools to calculate the bracket onHochschild cohomology in Section 2. Then, we compute the Gerstenhaber bracket onthe Hochschild cohomology of A = k [ x ] / ( x p ) where the field k has characteristic 0 andthe integer p > A for the case that k has positive characteristic p [7, Section 5].In Section 4, we compute the Gerstenhaber bracket for the Taft algebra T p whichis a nonquasi-triangular Hopf algebra. We use a similar technique as in [7] to calculatethe bracket on Hochschild cohomology of T p . It is also known that the Hopf algebracohomology of any Hopf algebra with a bijective antipode can be embedded in theHochschild cohomology of the algebra [14, Theorem 9.4.5 and Corollary 9.4.7]. Since allfinite dimensional Hopf algebras (also most of known infinite dimensional Hopf algebras)have bijective antipode, we can embed the Hopf algebra cohomology of T p into theHochschild cohomology of T p . Then, we use this explicit embedding and find the bracketon the Hopf algebra cohomology of T p . As a result of our calculation, we obtain thatthe bracket on Hopf algebra cohomology of T p is also trivial.In the last section, we derive a general expression for the bracket on Hopf algebracohomology of any Hopf algebra A with bijective antipode. We first consider a specificresolution that agrees with the bar resolution of A and find a bracket formula for it.Then, we use the composition of various isomorphisms and an embedding from Hopfalgebra cohomology into Hochschild cohomology in order to discover the bracket formulaon Hopf algebra cohomology. Let k be a field, A be a k -algebra, and A e = A ⊗ k A op where A op is the opposite algebrawith reverse multiplication. For simplicity, we write ⊗ instead of ⊗ k . The followingresolution B ( A ) is a free resolution of the A e -module A , called the bar resolution , B ( A ) : · · · d −→ A ⊗ d −→ A ⊗ d −→ A ⊗ π −→ A −→ , (2.1)where d n ( a ⊗ a ⊗ · · · ⊗ a n +1 ) = n (cid:88) i =0 ( − i a ⊗ a ⊗ · · · ⊗ a i a i +1 ⊗ · · · ⊗ a n +1 and π is multiplication.Consider the following complex that is derived by applying Hom A e ( − , A ) to the barresolution B ( A )0 −→ Hom A e ( A ⊗ , A ) d ∗ −→ Hom A e ( A ⊗ , A ) d ∗ −→ Hom A e ( A ⊗ , A ) d ∗ −→ · · · (2.2)2here d ∗ n ( f ) = f d n . The Hochshild cohomology of the algebra A is the cohomology ofthe cochain complex (2.1), i.e.HH ∗ ( A, A ) = (cid:77) n ≥ Ext nA e ( A, A ) . We also define the
Hopf algebra cohomology of the Hopf algebra A over the field k asH ∗ ( A, k ) = (cid:77) n ≥ Ext nA ( k, k )under the cup product.Let f ∈ Hom k ( A ⊗ m , A ) and g ∈ Hom k ( A ⊗ n , A ). The Hochschild cohomology of A is an algebra with the following cup product and the Gerstenhaber bracket structures.The cup product f (cid:94) g ∈ Hom k ( A ⊗ ( m + n ) , A ) is defined by( f (cid:94) g )( a ⊗ · · · ⊗ a m + n ) := ( − mn f ( a ⊗ · · · ⊗ a m ) g ( a m +1 ⊗ · · · a m + n )for all a , · · · , a m + n ∈ A , and the Gerstenhaber bracket [ f, g ] is an element ofHom k ( A ⊗ ( m + n − , A ) given by[ f, g ] := f ◦ g − ( − ( m − n − g ◦ f where the circle product f ◦ g is( f ◦ g )( a ⊗ · · · ⊗ a m + n − ) := m (cid:88) i =1 ( − ( n − i − f ( a ⊗ · · · a i − ⊗ g ( a i ⊗ · · · a i + n − ) ⊗ a i + n ⊗ · · · ⊗ a m + n − )for all a , · · · , a m + n − ∈ A . We note that these definitions directly come from the barresolution.There is an identity between cup product and bracket [3, Section 1]:[ f ∗ (cid:94) g ∗ , h ∗ ] = [ f ∗ , h ∗ ] (cid:94) g ∗ + ( − | f ∗ | ( | h ∗ |− f ∗ (cid:94) [ g ∗ , h ∗ ] , (2.3)where f ∗ , g ∗ , and h ∗ are the images (in Hochschild cohomology) of the cocyles f, g , and h , respectively.Computing the bracket on the bar resolution is not an ideal method. Instead, we canuse another resolution, A µ → A , satisfying the following hypotheses [7, (3.1) and Lemma3.4.1]:(a) A admits an embedding ι : A → B ( A ) of complexes of A -bimodules for which thefollowing diagram commutes A B ( A ) A ι ι admits a section π : B → A , i.e. an A e -chain map π with πι = id A .(c) There is a diagonal map that satisfies ∆ (2) A = ( π ⊗ A π ⊗ A π )∆ (2) B ( A ) ι where ∆ (2) =( id ⊗ ∆)∆.We give the following theorem which is the combination of [7, Theorem 3.2.5] and[7, Lemma 3.4.1] that allows us to use a different resolution for the bracket calculation. Theorem 2.4.
Suppose A µ → A is a projective A -bimodule resolution of A that satisfiesthe hypotheses (a)-(c). Let φ : A ⊗ A A → A be any contracting homotopy for the chainmap F A : A ⊗ A A → A defined by F A := ( µ ⊗ A id A − id A ⊗ A µ ) , i.e. d ( φ ) := d A φ + φd A ⊗ A A = F A . (2.5) Then for cocycles f and g in Hom A e ( A , A ) , the bracket given by [ f, g ] φ = f ◦ φ g − ( − ( | f |− | g |− g ◦ φ f (2.6) where the circle product is f ◦ φ g = f φ ( id A ⊗ A g ⊗ A id A )∆ (2) (2.7) agrees with the Gerstenhaber bracket on cohomology. In general, it is not easy to calculate the map φ by the formula (2.5). We usealternative way to find φ .Let h be any k -linear contracting homotopy for the identity map on the extendedcomplex A → A → A is free. A contracting homotopy φ i : ( A ⊗ A A ) i −→ A i +1 in Theorem 2.4 is constructed by the following formula [7, Lemma 3.3.1]: φ i = h i (( F A ) i − φ i − d ( A ⊗ A A ) i ) . (2.8) A = k [ x ] / ( x p ) Let A = k [ x ] / ( x p ) where k is a field of characteristic 0 and p > A by Theorem 2.4. We work on asmaller resolution of A than the bar resolution of A . Consider the following A e -moduleresolution of A : A : · · · v. −→ A e u. −→ A e v. −→ A e u. −→ A e π −→ A −→ , (3.1)where u = x ⊗ − ⊗ x , v = x p − ⊗ x p − ⊗ x + · · · + x ⊗ x p − + 1 ⊗ x p − , and π isthe multiplication.The bracket on A where k is a field with positive characteristic, is calculated by C.Negron and S. Witherspoon [7, Section 5]. We adopt the contracting homotopy h forthe identity map from that calculation and obtain a new map h for our setup. Let ξ i
4e the element 1 ⊗ A i . The following maps h n : A n −→ A n +1 form a contractinghomotopy for identity map, as we can see by direct calculation: h − ( x i ) = ξ x i ,h ( x i ξ x j ) = i − (cid:88) l =0 x l ξ x i + j − − l ,h ( x i ξ x j ) = δ i,p − x j ξ ,h n ( x i ξ n x j ) = − j − (cid:88) l =0 x i + j − − l ξ n +1 x l ( n ≥
2) , h n +1 ( x i ξ n +1 x j ) = δ j,p − x i ξ n +2 ( n ≥
2) . (3.2)Then, we take φ − = 0 and construct the following A e -linear maps φ i : ( A ⊗ A A ) i −→ A i +1 for degree 1 and 2 by (2.8): φ ( ξ ⊗ A x i ξ ) = i − (cid:88) l =0 x l ξ x i − − l ,φ ( ξ ⊗ A x i ξ ) = − δ i,p − ξ ,φ ( ξ ⊗ A x i ξ ) = δ i,p − ξ . (3.3)Lastly, we form the following diagonal map ∆ : A −→ A ⊗ A A :∆ ( ξ ) = ξ ⊗ A ξ , ∆ ( ξ ) = ξ ⊗ A ξ + ξ ⊗ A ξ , ∆ n ( ξ n ) = n (cid:88) i =0 ξ i ⊗ A ξ n − i + n − (cid:88) i =0 (cid:88) a + b + c = p − x a ξ i +1 ⊗ A x b ξ n − i − x c , for n ≥ n +1 ( ξ n +1 ) = n +1 (cid:88) i =0 ξ i ⊗ A ξ n +1 − i , for n ≥ . (3.4)It can be seen that the map ∆ is a chain map lifting the canonical isomorphism A ∼ → A ⊗ A A by direct calculation.Now, we are ready to calculate the brackets on cohomology in low degrees. Byapplying Hom A e ( − , A ) to A , we see that the differentials are all 0 in odd degrees and( px p − ) · in even degrees. In each degree, the term in the Hom complex is the free A -module Hom A e ( A e , A ) ∼ = A . Moreover, since p is not divisible by the characteristic of k ,we deduce HH ( A ) ∼ = A, HH i +1 ( A ) ∼ = ( x ) , and HH i ( A ) ∼ = A/ ( x p − ) [14, Section 1.1].Let x j ξ ∗ i ∈ Hom A e ( A e , A ) denote the function that takes ξ i to x j . Since the charac-teristic of k does not divide p , the Hochschild cohomology as an A -algebra is generatedby ξ ∗ and ξ ∗ [14, Example 2.2.2]. We only calculate the brackets of the elements of5egrees 1 and 2 which can be extended to higher degrees by the formula (2.3). Hence,we have the following calculations:The bracket of the elements of degrees 1 and 1:( x i ξ ∗ ◦ φ x j ξ ∗ )( ξ )= x i ξ ∗ φ (1 ⊗ A x j ξ ∗ ⊗ A (2) ( ξ )= x i ξ ∗ φ (1 ⊗ A x j ξ ∗ ⊗ A ξ ⊗ A ξ ⊗ A ξ + ξ ⊗ A ξ ⊗ A ξ + ξ ⊗ A ξ ⊗ ξ )= x i ξ ∗ φ ( ξ ⊗ A x j ξ )= x i ξ ∗ ( ξ x j − + xξ x j − + · · · + x j − ξ )= jx i + j − and by symmetry ( x j ξ ∗ ◦ φ x i ξ ∗ )( ξ ) = ix i + j − . Therefore, we have[ x i ξ ∗ , x j ξ ∗ ] = ( j − i ) x i + j − ξ ∗ . The bracket of the elements of degrees 1 and 2:( x i ξ ∗ ◦ φ x j ξ ∗ )( ξ )= x i ξ ∗ φ (1 ⊗ A x j ξ ∗ ⊗ A (2) ( ξ )= x i ξ ∗ φ (1 ⊗ A x j ξ ∗ ⊗ A ξ ⊗ A ξ ⊗ A ξ + ξ ⊗ A ξ ⊗ A ξ + ξ ⊗ A ξ ⊗ A ξ + ξ ⊗ A (cid:88) a + b + c = p − ( x a ξ ⊗ A x b ξ x c ) + (cid:88) a + b + c = p − x a ξ ⊗ A x b ( ξ ⊗ A ξ + ξ ⊗ A x ) x c )= x i ξ ∗ φ ( ξ ⊗ A x j ξ ) = x i ξ ∗ ( ξ x j − + xξ x j − + · · · + x j − ξ ) = jx i + j − . x j ξ ∗ ◦ φ x p − ξ ∗ )( ξ )= x j ξ ∗ φ (1 ⊗ A x p − ξ ∗ ⊗ A (2) ( ξ )= x j ξ ∗ φ (1 ⊗ A x p − ξ ∗ ⊗ A ξ ⊗ A ξ ⊗ A ξ + ξ ⊗ A ξ ⊗ A ξ + ξ ⊗ A ξ ⊗ A ξ + ξ ⊗ A (cid:88) a + b + c = p − ( x a ξ ⊗ A x b ξ x c ) + (cid:88) a + b + c = p − x a ξ ⊗ A x b ( ξ ⊗ A ξ + ξ ⊗ A x ) x c )= x j ξ ∗ φ ( (cid:88) a + b + c = p − ( ξ ⊗ A x a + b + i ξ x c − x a ξ ⊗ A x b + i ξ x c ))= x j ξ ∗ ( (cid:88) a + b + c = p − ( δ a + b + i,p − ξ x c + x a δ b + i,p − ξ x c ))= x j ξ ∗ (( p − i ) ξ x i − + (cid:88) a + c = i − x a ξ x c )= ( p − i ) x i + j − + (cid:88) a + c = i − x a + c + j = ( p − i ) x i + j − + ix i + j − = px i + j − . Therefore, we obtain [ x i ξ ∗ , x j ξ ∗ ] = ( j − p ) x i + j − ξ ∗ . Lastly, the bracket of the elements of degrees 2 and 2:( x i ξ ∗ ◦ φ x j ξ ∗ )( ξ ) = x i ξ ∗ φ (1 ⊗ A x j ξ ∗ ⊗ A (2) ( ξ )= x i ξ ∗ φ ( ξ ⊗ A x j ξ + ξ ⊗ A x j ξ ) = x i ξ ∗ (0) = 0and by symmetry ( x j ξ ∗ ◦ φ x i ξ ∗ )( ξ ) = 0. Therefore, we have[( x i ξ ∗ , x j ξ ∗ )] = 0 . As a consequence, the brackets for the elements of degrees 1 and 2 are[( x i ξ ∗ , x j ξ ∗ )] = ( j − i ) x i + j − ξ ∗ , [( x i ξ ∗ , x j ξ ∗ )] = ( j − p ) x i + j − ξ ∗ , [( x i ξ ∗ , x j ξ ∗ )] = 0 . Brackets in higher degrees can be determined from these and the identity (2.3) sincethe Hochschild cohomology is generated as an A -algebra under the cup product in degrees1 and 2.L. Grimley, V. C. Nguyen, and S. Witherspoon [4] calculated Gerstenhaber bracketson Hochschild cohomology of a twisted tensor product of algebras. S. Sanchez-Flores [9]7lso calculated the bracket on group algebras of a cyclic group over a field of positivecharacteristic which is isomorphic to A = k [ x ] / ( x p ). C. Negron and S. Witherspoon[7] calculated the bracket on group algebras of a cyclic group over a field of positivecharacteristic as well with the same h, φ , and ∆ maps. Our calculation agrees withthose except slightly different [( x i ξ ∗ , x j ξ ∗ )]. The Taft algebra T p with p > k -algebra generated by g and x satisfying therelations : g p = 1 , x p = 0 , and xg = ωgx where ω is a primitive p -th root of unity. It isa Hopf algebra with the structure: • ∆( g ) = g ⊗ g , ∆( x ) = 1 ⊗ x + x ⊗ g • ε ( g ) = 1 , ε ( x ) = 0 • S ( g ) = g − , S ( x ) = − xg − . Note that as an algebra, T p is a skew group algebra A (cid:111) kG where A = k [ x ] / ( x p ) and G = < g | g p = 1 > . The action of G on A is given by g x = ωx .In this section, our main goal is to calculate the bracket on Hochschild cohomology of T p with the same technique in Section 3 and find the bracket on Hopf algebra cohomologyof T p by using the embedding of H ∗ ( T p , k ) into HH ∗ ( T p , T p ).We first find the bracket on Hochschild cohomology of T p . Let D be the skew groupalgebra A e (cid:111) G where the action of G on A e is diagonal, i.e. g ( a ⊗ b ) = ( g a ) ⊗ ( g b ). Then,there is the following isomorphism [1, Section 2] D = A e (cid:111) G ∼ = (cid:77) g ∈ G Ag ⊗ Ag − ⊂ T ep . Hence D is isomorphic to a subalgebra of T ep via a ⊗ a ⊗ g (cid:55)→ a g ⊗ ( g − a g − ). Moreover, A is a D -module under the following left and right action [1, Section 4]:( a g ⊗ a g − ) a = a ga a g − = a ( g ( a a )) a ( a g ⊗ a g − ) = a g − a a g = a ( g − ( a a )) . Remember the resolution (3.1) A : · · · v. −→ A e u. −→ A e v. −→ A e u. −→ A e π −→ A −→ . This is also a D -projective resolution of A and the action of G on A e is given by • g · ( a ⊗ a ) = ( g a ) ⊗ ( g a ) in even degrees, • g · ( a ⊗ a ) = ω ( g a ) ⊗ ( g a ) in odd degrees.8rom the resolution A , we construct the following T ep resolution of T p : T ep ⊗ D A : · · · −→ T ep ⊗ D A e −→ T ep ⊗ D A e −→ T ep ⊗ D A e −→ T ep ⊗ D A −→ . (4.1)It is known that, T p ∼ = T ep ⊗ D A as T p -bimodules via the map sending x i ⊗ g k to (1 ⊗ g k ) ⊗ D x i [14, Section 3.5]. Then we have A ⊗ T p ∼ = T ep ⊗ D A e with the T p -bimoduleisomorphism given by κ ( x i ⊗ ( x j ⊗ g k )) = (1 ⊗ g k ) ⊗ D ( x i ⊗ x j ) . (4.2)Then, we obtain the following resolution ˜ A which is isomorphic to the resolution(4.1), i.e. ˜ A : · · · ˜ u. −→ A ⊗ T p ˜ v. −→ A ⊗ T p ˜ u. −→ A ⊗ T p ˜ π. −→ T p −→ v = v ⊗ id kG , ˜ u = u ⊗ id kG , and ˜ π = π ⊗ id kG .The following lemma gives us a contracting homotopy for the identity map on theresolution ˜ A . Lemma 4.4.
Let h n be a contracting homotopy in (3.2) . Then ˜ h n = h n ⊗ kG forms acontracting homotopy for the identity map on ˜ A .Proof. For n ≥
0, the domain of h n ⊗ kG is A ⊗ A ⊗ kG which is A ⊗ T p as a vectorspace. Moreover, by definition of contracting homotopy, h n satisfy h i − d i + d i +1 h i = id A i . Then,˜ h i − ˜ d i + ˜ d i +1 ˜ h i = ( h i − ⊗ id kG )( d i ⊗ id kG ) + ( d i +1 ⊗ id kG )( h i ⊗ id kG )= ( h i − d i ⊗ id kG ) + ( d i +1 h i ⊗ id kG ) = ( h i − d i + d i +1 h i ) ⊗ id kG = id A i ⊗ id kG = id ˜ A i and that implies ˜ h n is a contracting homotopy for ˜ A . The proof is similar for n = − a ⊗ a ⊗ g ∈ A ⊗ T p by a ⊗ a g . By the Lemma 4.4, we obtain˜ h − ( x i g ) = ξ x i g, ˜ h ( x i ξ x j g ) = i − (cid:88) l =0 x l ξ x i + j − − l g, ˜ h ( x i ξ x j g ) = δ i,p − x j ξ g, ˜ h n ( x i ξ n x j g ) = − j − (cid:88) l =0 x i + j − − l ξ n +1 x l g, ˜ h n +1 ( x i ξ n +1 x j g ) = δ j,p − x i ξ n +2 g.
9e need a lemma to have the linear maps ˜ φ i : ( ˜ A ⊗ T p ˜ A ) i −→ ˜ A i +1 . However, we firstmention that there is an isomorphism from ( A ⊗ T p ) ⊗ T p ( A ⊗ T p ) to ( A ⊗ A ) ⊗ A ( A ⊗ A ) ⊗ kG as T ep -modules given by ψ (( x i ⊗ x j g k ) ⊗ T p ( x i ⊗ x j g k )) = ω k ( i + j ) ( x i ⊗ x j ) ⊗ A ( x i ⊗ x j ) g ( k + k ) . (4.5) Lemma 4.6.
Let F A = ( π ⊗ A id A − id A ⊗ A π ) be the chain map for the resolution A in (3.1) which is used for calculation of φ in (3.3) . Then F ˜ A : ˜ A ⊗ T p ˜ A → ˜ A definedby (˜ π ⊗ T p id ˜ A − id ˜ A ⊗ T p ˜ π ) is exactly ( F A ⊗ id kG ) ψ . Moreover ˜ φ := ( φ ⊗ id kG ) ψ is acontracting homotopy for F ˜ A .Proof. Let ( x i ⊗ x j g k ) ⊗ T p ( x i ⊗ x j g k ) ∈ ( A ⊗ T p ) ⊗ T p ( A ⊗ T p ). Note that F ˜ A is zeroif degrees of ( x i ⊗ x j g k ) and ( x i ⊗ x j g k ) are both nonzero since ˜ π is only defined ondegree zero. Also remember that ˜ π = π ⊗ id kG for the resolution ˜ A .We check the case that the degree of ( x i ⊗ x j g k ) is zero and the degree of ( x i ⊗ x j g k ) is nonzero. By using definition of F ˜ A , we obtain F ˜ A (( x i ⊗ x j g k ) ⊗ T p ( x i ⊗ x j g k )) = ( x i + j g k ) ⊗ T p ( x i ⊗ x j g k )= ω k ( i + j ) x i + i + j ⊗ x i g k + k . On the other hand, we also have( F A ⊗ id kG ) ψ (( x i ⊗ x j g k ) ⊗ T p ( x i ⊗ x j g k ))= ( F A ⊗ id kG )( ω k ( i + j ) ( x i ⊗ x j ) ⊗ A ( x i ⊗ x j ) g k + k )= ω k ( i + j ) x i + i + j ⊗ x i g k + k . The proof for other cases are similar. Hence F ˜ A and ( F A ⊗ id kG ) ψ are identical.In order to prove ˜ φ := ( φ ⊗ id kG ) ψ is a contracting homotopy for F ˜ A , we need toshow that ˜ d ˜ A ˜ φ + ˜ φ ˜ d ˜ A ⊗ Tp ˜ A = F ˜ A . It is clear that ˜ d ˜ A ˜ φ = ( d A ⊗ id kG )( φ ⊗ id kG ) ψ = ( d A φ ⊗ id kG ) ψ. (4.7)We now claim that ψ ˜ d ˜ A ⊗ Tp ˜ A = ( d A ⊗ A A ⊗ id kG ) ψ. (4.8)By definition ˜ d ˜ A ⊗ Tp ˜ A = ˜ d ˜ A ⊗ T p id T p + ( − ∗ id T p ⊗ T p ˜ d ˜ A where ∗ is the degree of the element in left A ⊗ T p . Moreover, ( A ⊗ T p ) ⊗ T p ( A ⊗ T p )is generated by ξ m G ⊗ T p x i ξ n G as T p -bimodule. Without loss of generality, assume m and n are odd. Then we have the following calculation: ψ ˜ d ˜ A ⊗ Tp ˜ A ( ξ m G ⊗ T p x i ξ n G )= ψ (( xξ m G − ξ m x G ) ⊗ T p x i ξ n G − ξ m G ⊗ T p ( x i +1 ξ n G − x i ξ n x G ))= ( xξ m − ξ m x ) ⊗ A x i ξ n G − ξ m ⊗ A ( x i +1 ξ n − x i ξ n x )1 G d A ⊗ A A ⊗ id kG ) ψ ( ξ m G ⊗ T p x i ξ n G )= ( d A ⊗ A A ⊗ id kG )( ξ m ⊗ A x i ξ n G )= ( xξ m − ξ m x ) ⊗ A x i ξ n G − ξ m ⊗ A ( x i +1 ξ n − x i ξ n x )1 G . The calculation is similar for the other cases of m and n . Therefore,˜ φ ˜ d ˜ A ⊗ Tp ˜ A = ( φ ⊗ id kG ) ψ ˜ d ˜ A ⊗ Tp ˜ A = ( φ ⊗ id kG )( d A ⊗ A A ⊗ id kG ) ψ = ( φd A ⊗ A A ⊗ id kG ) ψ. (4.9)By combining (4.7) and (4.9), we obtain˜ d ˜ A ˜ φ + ˜ φ ˜ d ˜ A ⊗ Tp ˜ A = (( d A φ + φd A ⊗ A A ) ⊗ id kG ) ψ = ( F A ⊗ id kG ) ψ = F ˜ A whence ˜ φ = ( φ ⊗ id kG ) ψ is a contracting homotopy for F ˜ A .We use the Lemma 4.6 and find the following T ep -linear maps ˜ φ i : ( ˜ A ⊗ T p ˜ A ) i −→ ˜ A i +1 :˜ φ ( ξ G ⊗ T p x i ξ G ) = i − (cid:88) l =0 x l ξ x i − − l G , ˜ φ ( ξ G ⊗ T p x i ξ G ) = − δ i,p − ξ G , ˜ φ ( ξ G ⊗ T p x i ξ G ) = δ i,p − ξ G . Next, we give a lemma to find the the diagonal map.
Lemma 4.10.
The map ˜∆ := ψ − (∆ ⊗ id kG ) is a diagonal map on ˜ A where ∆ is in (3.4) .Proof. We need to check that ˜∆ is a chain map. The following equations are straight-forward by considering the fact that ∆ is a chain map and (4.8):˜ d ˜ A ⊗ Tp ˜ A ˜∆ = ˜ d ˜ A ⊗ Tp ˜ A ψ − (∆ ⊗ id kG ) = ψ − ( d A ⊗ A A ⊗ id kG )(∆ ⊗ id kG )= ψ − ( d A ⊗ A A ∆ ⊗ id kG ) = ψ − (∆ d A ⊗ id kG ) = ψ − (∆ ⊗ id kG )( d A ⊗ id kG )= ˜∆ ˜ d ˜ A . Lemma 4.10 allows us to compute the T p -linear map ˜∆ : ˜ A i +1 −→ ( ˜ A ⊗ T p ˜ A ) i as11ollows: ˜∆ ( ξ G ) = ξ G ⊗ T p ξ G , ˜∆ ( ξ G ) = ξ G ⊗ T p ξ G + ξ G ⊗ T p ξ G , ˜∆ n ( ξ n G ) = n (cid:88) i =0 ξ i G ⊗ T p ξ n − i G + n − (cid:88) i =0 (cid:88) a + b + c = p − x a ξ i +1 G ⊗ T p x b ξ n − i − x c G , for n ≥ n +1 ( ξ n +1 G ) = n +1 (cid:88) i =0 ξ i G ⊗ T p ξ n +1 − i G , for n ≥ . Before computing the bracket on Hochschild cohomology of T p , we need to find abasis of Hom T ep ( ˜ A , T p ). In particular, we must find a basis of Hom T ep ( A ⊗ T p , T p ) as it isan invariant in each degree.It is known thatHH ∗ ( T p ) := Ext ∗ T ep ( T p , T p ) ∼ = Ext ∗D ( A, T p ) ∼ = Ext ∗ A e ( A, T p ) G . The Eckmann-Shapiro Lemma (Lemma 5.3) and (4.2) imply the first isomorphism andsee [14, Theorem 3.6.2] for the second isomorphism.Consider the following resolutionHom A e ( A , T p ) G : 0 −→ Hom A e ( A e , T p ) G −→ Hom A e ( A e , T p ) G −→ · · · (4.11)where the action of G on Hom A e ( A e , T p ) G is defined by g · f ( a ⊗ a ) = g f ( g − ( a ⊗ a )) . (4.12)This resolution is clearly isomorphic to0 −→ T Gp −→ T Gp −→ T Gp −→ · · · (4.13)with the correspondence f t (cid:55)→ t where f t ( ξ ∗ ) = t for all t ∈ T p . (4.14)We claim that Hom T ep ( A ⊗ T p , T p ) ∼ = T Gp . Suppose x i g j ∈ T Gp . Then, we have f x i g j ∈ Hom A e ( A e , T p ) G defined by f x i g j ( x k ⊗ x l ) := x k + l + i g j where x ∗ ∈ A . Now observe that, f x i g j ∈ Hom A e ( A e , T p ) G is a D -module homomorphism since f x i g j (( x k ξ ∗ x l g )( a ⊗ a )) = f x i g j (( x k ξ ∗ x l G ) g ( a ⊗ a )) = ( x k ξ ∗ x l G ) f x i g j ( g ( a ⊗ a ))= ( x k ξ ∗ x l G ) gf x i g j ( a ⊗ a ) = ( x k ξ ∗ x l g ) f x i g j ( a ⊗ a )12here x k ξ ∗ x l g ∈ D , a ⊗ a ∈ A e . Moreover, if f ∈ Hom D ( A e , T p ), then f is G -invariantas g · f ( a ⊗ a ) = g f ( g − ( a ⊗ a )) = ( gg − ) f ( a ⊗ a ) = f ( a ⊗ a )where g ∈ G, a ⊗ a ∈ A e . Hence, the isomorphism from Hom A e ( A e , T p ) G to Hom D ( A e , T p )is the identity, so that f x i g j is also in Hom D ( A e , T p ). We next use the Eckmann-Shapirolemma (Lemma 5.3) which implies that Ext ∗D ( A, T p ) ∼ = Ext ∗ T ep ( T ep ⊗ D A, T p ) and theisomorphism is given by σ ( f x i g j )( x m g s ⊗ x n g r ⊗ D x k ⊗ x l ) = x m g s ⊗ x n g r f x i g j ( x k ⊗ x l ) = x m g s ⊗ x n g r ( x k + l + i g j )= ( x m g s )( x k + l + i g j )( x n g r )= (( x m ( g s x k + l + i )) g s + j )( x n g r )= ω s ( k + l + i ) ( x m + k + l + i g s + j )( x n g r )= ω s ( k + l + i ) ( x m + k + l + i ( g s + j x n )) g j + s + r = ω s ( k + l + i + n )+ jn x i + k + l + m + n g j + s + r . Hence, σ ( f x i g j ) is in Hom T ep ( T ep ⊗ D A e , T p ). Lastly, recall that T ep ⊗ D A e ∼ = A ⊗ T p via κ (4.2); so that, κ ∗ ( σ ( f x i g j ))( x k ⊗ x l g r ) = σ ( f x i g j )((1 T p ⊗ ξ ∗ g r ) ⊗ D x k ⊗ x l ) = x i + k + l g j + r which implies κ ∗ ( σ ( f x i g j )) ∈ Hom T ep ( A ⊗ T p , T p ). For simplicity, we define ˜ f x i g j := κ ∗ ( σ ( f x i g j )).The action of G on T p given by (4.12) and (4.14) depends on degree. Since T Gp isspanned by { , g, · · · , g p − } in even degrees and { x, xg, · · · , xg p − } in odd degrees [8,Section 8.2], we have { ˜ f , ˜ f g , · · · , ˜ f g p − } in even degrees and { ˜ f x , ˜ f xg , · · · , ˜ f xg p − } in theodd degrees as a basis of Hom T ep ( A ⊗ T p , T p ).We only calculate the bracket in degree 1 and 2 as before so we can extend it to higherdegrees by the relation between cup product and the bracket. Since A ⊗ T p ∼ = A e ⊗ kG as vector spaces, ξ i G generates A ⊗ T p as a T p -bimodule. Through the calculation, id represents id A ⊗ T p and ⊗ represents ⊗ T p .The circle product of two elements in degree one is( ˜ f xg i ◦ ˜ φ ˜ f xg j )( ξ G ) = ˜ f xg i ˜ φ ( id ⊗ ˜ f xg j ⊗ id ) ˜∆ (2) ( ξ G )= ˜ f xg i ˜ φ ( id ⊗ ˜ f xg j ⊗ id )( ξ G ⊗ ξ G ⊗ ξ G + ξ G ⊗ ξ G ⊗ ξ G + ξ G ⊗ ξ G ⊗ ξ G )= ˜ f xg i ˜ φ ( ξ G ⊗ xξ g j ) = ˜ f xg i ( ξ g j ) = xg i + j . Because of the symmetry, ( ˜ f xg j ◦ ˜ φ ˜ f xg i )( ξ G ) = xg i + j . Therefore[ ˜ f xg i , ˜ f xg j ]( ξ G ) = xg i + j − ( − xg i + j = 0 . f xg i ◦ ˜ φ ˜ f g j )( ξ G ) = ˜ f xg i ˜ φ ( id ⊗ ˜ f g j ⊗ id ) ˜∆ (2) ( ξ G ) = ˜ f xg i ˜ φ ( id ⊗ ˜ f g j ⊗ id )( ξ G ⊗ ξ G ⊗ ξ G + ξ G ⊗ ξ G ⊗ ξ G + ξ G ⊗ (cid:88) a + b + c = p − ( x a ξ G ⊗ x b ξ x c G ) + ξ G ⊗ ξ G ⊗ ξ G + (cid:88) a + b + c = p − ( x a ξ G ⊗ ( x b ξ G ⊗ ξ x c G + x b ξ G ⊗ ξ x c G )))= ˜ f xg i ˜ φ ( ξ G ⊗ ξ g j ) = 0 . And the circle product on the reverse order:( ˜ f g j ◦ ˜ φ ˜ f xg i )( ξ G ) = ˜ f g j ˜ φ ( id ⊗ ˜ f xg i ⊗ id ) ˜∆ (2) ( ξ G ) = ˜ f g j ˜ φ ( id ⊗ ˜ f xg i ⊗ id )( ξ G ⊗ ξ G ⊗ ξ G + ξ G ⊗ ξ G ⊗ ξ G + ξ G ⊗ (cid:88) a + b + c = p − ( x a ξ G ⊗ x b ξ x c G ) + ξ G ⊗ ξ G ⊗ ξ G + (cid:88) a + b + c = p − ( x a ξ G ⊗ ( x b ξ G ⊗ ξ x c G + x b ξ G ⊗ ξ x c G )))= ˜ f g j ˜ φ ( (cid:88) a + b + c = p − ω i ( b + c ) ξ G ⊗ x a + b +1 ξ x c g i + ω ic x a ξ G ⊗ x b +1 ξ x c g i )= ˜ f g j ( (cid:88) a + b + c = p − ω i ( b + c ) δ a + b +1 ,p − x c ξ g i − ω ic δ b +1 ,p − x a + c ξ g i )= ˜ f g j ( p − (cid:88) b =0 ω ib ξ g i ) − ˜ f g j ( ξ g i )= (cid:26) ( p − g j , for i = 0 − ( ω − i + 1) g i + j , for i (cid:54) = 0 . Therefore, we obtain [ ˜ f xg i , ˜ f g j ] = (cid:26) − ( p − g j , for i = 0( ω − i + 1) g i + j , for i (cid:54) = 0 . Lastly, the bracket of the elements of degrees 2 and 2:14 ˜ f g i ◦ ˜ φ ˜ f g j )( ξ G ) = ˜ f g i ˜ φ ( id ⊗ ˜ f g j ⊗ id ) ˜∆ (2) ( ξ G ) = ˜ f g i ˜ φ ( id ⊗ ˜ f g j ⊗ id )( ξ G ⊗ ξ G ⊗ ξ G + ξ G ⊗ ξ G ⊗ ξ G + ξ G ⊗ ξ G ⊗ ξ G + ξ G ⊗ ξ G ⊗ ξ G + ξ G ⊗ ξ G ⊗ ξ G + ξ G ⊗ ξ G ⊗ ξ G + ξ G ⊗ ξ G ⊗ ξ G + ξ G ⊗ ξ G ⊗ ξ G + ξ G ⊗ ξ G ⊗ ξ G )= ˜ f g i ˜ φ ( ξ G ⊗ ξ g j + ξ G ⊗ ξ g j ) = 0and by symmetry ( ˜ f g j ◦ ˜ φ ˜ f g i )( ξ G ) = 0. Therefore, we have [ ˜ f g i , ˜ f g j ] = 0. As aconsequence, the bracket for the elements of degree 1 and 2 are[ ˜ f xg i , ˜ f xg j ] = 0 , [ ˜ f xg i , ˜ f g j ] = (cid:26) − ( p − g j , for i = 0( ω − i + 1) g i + j , for i (cid:54) = 0 , [ ˜ f g i , ˜ f g j ] = 0 . By the identity (2.3), brackets in higher degrees can be determined, since the Hochschildcohomology is generated as an algebra under cup product in degrees 1 and 2.Hopf algebra cohomology of T p and Hochschild cohomology of T p were calculatedbefore by V. C. Nguyen [8, Section 8] as the Hopf algebra cohomologyH n ( T p , k ) = (cid:40) k if n is even,0 if n is odd,and the Hochschild cohomologyHH n ( T p , k ) = (cid:40) k if n is even, Span k { x } if n is odd.It is known that for any Hopf algebra with bijective antipode, the Hopf algebracohomology can be embedded into the Hochschild cohomology [14, Theorem 9.4.5 andCorollary 9.4.7]. Since any finite dimensional Hopf algebra has a bijective antipode, theTaft algebra T p is also a Hopf algebra with a bijective antipode. The embedding ofH n ( T p , k ) into HH n ( T p , T p ) turns out to be the map that is identity in even degrees andzero on odd degrees. Then, the corresponding bracket in Hopf algebra cohomology is[ ˜ f g i , ˜ f g j ] = 0 , so that, the bracket on Hopf algebra cohomology for the elements of all degrees is 0 bythe identity (2.3).This is the first example of the Gerstenhaber bracket on the Hopf algebra cohomologyof a nonquasi-triangular Hopf algebra and our calculation shows that the bracket on Hopfalgebra cohomology of a Taft algebra is zero as it is on the Hopf algebra cohomologyof any quasi-triangular algebra. A natural question that arises whether the bracketstructure on the Hopf algebra cohomology is always trivial. In the next section, weexplore a general expression for the bracket on the Hopf algebra cohomology that mayhelp us to approach this question with a more theoretical perspective in the futureresearches. 15 Gerstenhaber bracket for Hopf algebras
In this section, we want to explore an expression for Gerstenhaber bracket on a Hopfalgebra A with a bijective antipode S .We give the following lemma which helps us to define the Gerstenhaber bracket onan equivalent resolution to the bar resolution of A as an A -bimodule. Lemma 5.1.
Let A be a Hopf algebra with bijective antipode. Let P • be the bar resolutionof k as a left A -module: P • : · · · d −→ A ⊗ d −→ A ⊗ d −→ A ε −→ k −→ , with differentials d n ( a ⊗ a ⊗· · ·⊗ a n ) = n − (cid:88) i =0 ( − i a ⊗ a ⊗· · ·⊗ a i a i +1 ⊗· · ·⊗ a n +( − n ε ( a n ) a ⊗· · ·⊗ a n − Then X • = A e ⊗ A P • is equivalent to the bar resolution of A as an A -bimodule.Proof. Since S is bijective [14, Lemma 9.2.9], A e is projective as a right A -module. Alsothere is an A e -module isomorphism ρ : A → A e ⊗ A k defined by ρ ( a ) = a ⊗ ⊗ a ∈ A [14, Lemma 9.4.2].For each n , define θ n : X n → A ⊗ ( n +2) by θ n (( a ⊗ b ) ⊗ A (1 ⊗ c ⊗ c ⊗ · · · ⊗ c n )) = (cid:88) a ⊗ c ⊗ c ⊗ · · · ⊗ c n ⊗ S ( c c · · · c n ) b for all a, b, c , · · · c n ∈ A. Now, we show that θ is a chain map: θ n − d n (( a ⊗ b ) ⊗ A (1 ⊗ c ⊗ c ⊗ · · · ⊗ c n ))= θ n − (( a ⊗ b ) ⊗ A ( c ⊗ c ⊗ · · · ⊗ c n )+ n − (cid:88) i =1 ( − i ( a ⊗ b ) ⊗ A (1 ⊗ c ⊗ c ⊗ · · · ⊗ c i c i +1 ⊗ · · · ⊗ c n )+ ( − n ( a ⊗ b ) ⊗ A ( ε ( c n ) ⊗ c ⊗ c ⊗ · · · ⊗ c n − ))= θ n − ( (cid:88) ( ac ⊗ S ( c ) b ) ⊗ A (1 ⊗ c ⊗ · · · ⊗ c n )+ n − (cid:88) i =1 ( − i ( a ⊗ b ) ⊗ A (1 ⊗ c ⊗ c ⊗ · · · ⊗ c i c i +1 ⊗ · · · ⊗ c n )+ ( − n ( ε ( c n ) a ⊗ b ) ⊗ A (1 ⊗ c ⊗ c ⊗ · · · ⊗ c n − ))= (cid:88) ac ⊗ c ⊗ · · · ⊗ c n ⊗ S ( c · · · c n ) S ( c ) b + n − (cid:88) i =1 ( − i (cid:88) a ⊗ c ⊗ · · · ⊗ c i c i +11 ⊗ · · · ⊗ c n ⊗ S ( c · · · c n ) b + (cid:88) ( − n a ⊗ c ⊗ · · · ⊗ c n − ⊗ ε ( c n ) S ( c · · · c n − ) b d n θ n (( a ⊗ b ) ⊗ A (1 ⊗ c ⊗ c ⊗ · · · ⊗ c n ))= d n ( (cid:88) a ⊗ c ⊗ c ⊗ · · · ⊗ c n ⊗ S ( c c · · · c n ) b )= (cid:88) ac ⊗ c ⊗ · · · ⊗ c n ⊗ S ( c c · · · c n ) b + (cid:88) n − (cid:88) i =1 ( − i a ⊗ c ⊗ · · · ⊗ c i c i +11 ⊗ · · · ⊗ c n ⊗ S ( c · · · c n ) b + (cid:88) ( − n a ⊗ c ⊗ · · · ⊗ c n − ⊗ c n S ( c · · · c n ) b. Since S is an algebra anti-homomorphism that is convolution inverse to the identitymap, (cid:88) c n S ( c · · · c n ) = (cid:88) c n S ( c n ) S ( c n − ) · · · S ( c ) = (cid:88) ε ( c n ) S ( c · · · c n − )and S ( c · · · c n ) S ( c ) = S ( c c · · · c n )so that the two expressions are equal which follows θ is a chain map.Lastly, one can see that the A e -module homomorphism ψ n ( a ⊗ c ⊗ c ⊗ · · · c n ⊗ b ) = (cid:88) ( a ⊗ c c · · · c n b ) ⊗ A (1 ⊗ c ⊗ c ⊗ · · · ⊗ c n )is the inverse of θ n by using the property that S is an algebra anti-homomorphism thatis convolution inverse to the identity map.Let f x ∈ Hom A e ( X m , A ) and g x ∈ Hom A e ( X n , A ). Then we define the X -bracket[ f x , g x ] X ∈ Hom A e ( X m + n − , A ) to be a composition X θ −→ B ( A ) [ ψ ∗ f x .ψ ∗ g x ] −−−−−−−→ A ; so that,we have[ f x , g x ] X = [ ψ ∗ f x , ψ ∗ g x ] θ = ( ψ ∗ f x ◦ ψ ∗ g x ) θ − ( − ( m − n − ( ψ ∗ g x ◦ ψ ∗ f x ) θ ψ ∗ f x ◦ ψ ∗ g x ) θ m + n − (( a ⊗ b ) ⊗ A ⊗ c ⊗ · · · ⊗ c m + n − )=( ψ ∗ f x ◦ ψ ∗ g x )( (cid:88) a ⊗ c ⊗ c ⊗ · · · ⊗ c m + n − ⊗ S ( c c · · · c m + n − ) b )= (cid:88) m (cid:88) i =1 ( − ( n − i − f x ψ m ( a ⊗ c ⊗ · · · ⊗ c i − ⊗ g x ψ n (1 ⊗ c i ⊗ · · · ⊗ c i + n − ⊗ ⊗ c i + n ⊗ · · · ⊗ c m + n − ⊗ S ( c c · · · c m + n − ) b )= (cid:88) m (cid:88) i =1 ( − ( n − i − f x ψ m ( a ⊗ c ⊗ · · · ⊗ c i − ⊗ (cid:88) g x (1 ⊗ c i c i +12 · · · c i + n − ⊗ A ⊗ c i ⊗ c i +11 ⊗ · · · ⊗ c i + n − ) ⊗ c i + n ⊗ · · · ⊗ c m + n − ⊗ S ( c · · · c i − c i · · · c i + n − c i + n · · · c m + n − ) b )= (cid:88) m (cid:88) i =1 ( − ( n − i − f x ( a ⊗ c c · · · c i − c ∗ c i + n · · · c m + n − S ( c c · · · c m + n − ) b ⊗ A ⊗ c ⊗ c ⊗ · · · ⊗ c i − ⊗ c ∗ ⊗ c i + n ⊗ · · · ⊗ c m + n − )where∆( c ) = (cid:88) c ⊗ c , ∆ (2) ( c ) = (cid:88) c ⊗ c ⊗ c , ∆( c ∗ ) = (cid:88) c ∗ ⊗ c ∗ and c ∗ = (cid:88) g x (1 ⊗ c i c i +12 · · · c i + n − ⊗ A ⊗ c i ⊗ c i +11 ⊗ · · · ⊗ c i + n − ) . This is the general expression of the Gerstenhaber bracket on Hochschild cohomologyof A . Next, we start with the following theorem [14, Theorem 9.4.5] to construct anembedding from H ∗ ( A, k ) into HH ∗ ( A ). Theorem 5.2.
Let A be a Hopf algebra over k with bijective antipode. Then HH ∗ ( A ) ∼ = H ∗ ( A, A ad ) . In this theorem A ad is an A -module A under left adjoint action, given by a · b = (cid:80) a bS ( a ) for all a, b ∈ A . To find explicit isomorphism between HH ∗ ( A ) andH ∗ ( A, A ad ), we give the Eckmann-Shapiro lemma. Lemma 5.3 (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 ) . Proof.
Let P • → N be a B projective resolution of N . Then A ⊗ B P n is projectiveas A-module so that A ⊗ B P • → A ⊗ B N is a projective resolution of A ⊗ B N as an A -module. Let σ : Hom B ( P n , M ) → Hom A ( A ⊗ B P n , M ) defined by σ ( f )( a ⊗ B p ) = af ( p ) , : Hom A ( A ⊗ B P n , M ) → Hom B ( P n , M ) defined by τ ( g )( p ) = g (1 ⊗ B p )where a ∈ A, p ∈ P n , f ∈ Hom B ( P n , M ) , g ∈ Hom A ( A ⊗ B P n , M ) . Since σ and τ are in-verse of each other and they are homomorphisms, Hom A ( A ⊗ B P n , M ) ∼ = Hom B ( P n , M ).If we replace A with A e , B with A and take M = A, N = k in the Eckmann-Shapirolemma, we have the isomorphism Ext nA e ( A e ⊗ A k, A ) ∼ =Ext nA ( k, A ad ). We also know that A ∼ = A e ⊗ A k [14, Lemma 9.4.2] and the isomorphism is given by ρ ( a ) = a ⊗ ⊗ a ∈ A . Therefore Ext nA e ( A, A ) ∼ =Ext nA e ( A e ⊗ A k, A ) ∼ =Ext nA ( k, A ad ).We already have the Gerstenhaber bracket [ , ] X on Ext nA e ( A e ⊗ A k, A ). Hence wecan use the isomorphisms σ and τ in Eckmann-Shapiro Lemma and find the bracketexpression on H ∗ ( A, A ad ). Now let ˜ f ∈ Hom A ( P m , A ad ) and ˜ g ∈ Hom A ( P n , A ad ). Then[ ˜ f , ˜ g ] P ∈ Hom A ( P m + n − , A ad ) and we have[ ˜ f , ˜ g ] P = τ [ σ ( ˜ f ) , σ (˜ g )] X = τ (( ψ ∗ ( σ ( ˜ f )) ◦ ψ ∗ ( σ (˜ g ))) θ ) − ( − ( m − n − τ (( ψ ∗ ( σ (˜ g )) ◦ ψ ∗ ( σ ( ˜ f ))) θ ) . For simplification we define˜ f ◦ P ˜ g := τ (( ψ ∗ ( σ ( ˜ f )) ◦ ψ ∗ ( σ (˜ g ))) θ ) . Then by using previous circle product formula we obtain:˜ f ◦ P ˜ g (1 ⊗ c ⊗ c ⊗ · · · ⊗ c m + n − )= τ (( ψ ∗ ( σ ( ˜ f )) ◦ ψ ∗ ( σ (˜ g ))) θ )(1 ⊗ c ⊗ c ⊗ · · · ⊗ c m + n − )=( ψ ∗ ( σ ( ˜ f )) ◦ ψ ∗ ( σ (˜ g ))) θ ((1 ⊗ ⊗ A ⊗ c ⊗ c ⊗ · · · ⊗ c m + n − )= (cid:88) m (cid:88) i =1 ( − ( n − i − σ ( ˜ f )(1 ⊗ c c · · · c i − c ∗ c i + n · · · c m + n − S ( c c · · · c m + n − ) ⊗ A ⊗ c ⊗ c ⊗ · · · ⊗ c i − ⊗ c ∗ ⊗ c i + n ⊗ · · · ⊗ c m + n − )= (cid:88) m (cid:88) i =1 ( − ( n − i − ˜ f (1 ⊗ c ⊗ c ⊗ · · · ⊗ c i − ⊗ c ∗ ⊗ c i + n ⊗ · · · ⊗ c m + n − ) c c · · · c i − c ∗ c i + n · · · c m + n − S ( c c · · · c m + n − ))with ∆( c ∗ ) = (cid:80) c ∗ ⊗ c ∗ and c ∗ = (cid:88) σ (˜ g )(1 ⊗ c i c i +12 · · · c i + n − ⊗ A ⊗ c i ⊗ c i +11 ⊗ · · · ⊗ c i + n − )= (cid:88) (1 ⊗ c i c i +12 · · · c i + n − )˜ g (1 ⊗ c i ⊗ c i +11 ⊗ · · · ⊗ c i + n − )= (cid:88) ˜ g (1 ⊗ c i ⊗ c i +11 ⊗ · · · ⊗ c i + n − ) c i c i +12 · · · c i + n − .
19e now have the Lie bracket [ , ] P on H ∗ ( A, A ad ). Next, we embed H ∗ ( A, k ) intoH ∗ ( A, A ad ) [14, Corollary 9.4.7] via the unit map η ∗ : Hom A ( P • , k ) → Hom A ( P • , A ad ) . Let f ∈ Hom A ( P m , k ) and g ∈ Hom A ( P n , k ). Then by using counit map ε ∗ : Hom A ( P • , A ) → Hom A ( P • , k ) ,η ∗ and bracket on H ∗ ( A, A ad ), we derive the formula for [ f, g ] ∈ Hom A ( P m + n − , k ):[ f, g ] = ε ∗ [ η ∗ ( f ) , η ∗ ( g )] P = ε ∗ ( η ∗ ( f ) ◦ P η ∗ ( g )) − ( − ( m − n − ε ∗ ( η ∗ ( g ) ◦ P η ∗ ( f ))where ε ∗ (( η ∗ ( f ) ◦ P η ∗ ( g ))(1 ⊗ c ⊗ c ⊗ · · · ⊗ c m + n − ))= ε ( (cid:88) m (cid:88) i =1 ( − ( n − i − η ( f (1 ⊗ c ⊗ c ⊗ · · · ⊗ c i − ⊗ c ∗ ⊗ c i + n ⊗ · · · ⊗ c m + n − )) c c · · · c i − c ∗ c i + n · · · c m + n − S ( c c · · · c m + n − ))with ∆( c ∗ ) = (cid:88) c ∗ ⊗ c ∗ and c ∗ = (cid:88) η ( g (1 ⊗ c i ⊗ c i +11 ⊗ · · · ⊗ c i + n − )) c i c i +12 · · · c i + n − . Therefore, the last formula is a general expression of the Gerstenhaber bracket on aHopf algebra cohomology which is indeed inherited from the formula of the bracket onHochschild cohomology.
Acknowledgement
The author would like to thank S. Witherspoon for her precious time, suggestions andsupport.
References [1]
S. M. Burciu and S. Witherspoon , Hochschild cohomology of smash productsand rank one Hopf algebras , Biblioteca de la Revista Matematica IberoamericanaActas del ”XVI Coloquio Latinoamericano de Algebra” (2005), 153-170.[2]
M.A. Farinati, A. Solotar , G-structure on the cohomology of Hopf algebras ,Proceedings of the American Mathematical Society 132 (2004), no. 10, 2859–2865.[3]
M. Gerstenhaber , The cohomology structure of an associative ring , Ann. of Math.(2) (1963), 78:267-288. 204]
L. Grimley, V. C. Nguyen and S. Witherspoon , Gerstenhaber brackets onHochschild cohomology of twisted tensor products , J. Noncommutative Geometry 11(2017), no. 4, 1351–1379.[5]
R. Hermann , Monoidal Categories and the Gerstenhaber Bracket in HochschildCohomology , American Mathematical Society, 2016.[6]
G. Hochschild , On the cohomology groups of an associative algebra , Ann. Math.(2) (1945), 58-67.[7]
C. Negron and S. Witherspoon , An alternate approach to the Lie bracket onHochschild cohomology , Homology, Homotopy and Applications 18 (2016), no.1,265-285.[8]
V. C. Nguyen , Tate and Tate-Hochschild cohomology for finite dimensional Hopfalgebras , J. Pure Appl. Algebra, 217 (2013), 1967-1979.[9]
S. S´anchez-Flores , The Lie structure on the Hochschild cohomology of a modulargroup algebra , J. Pure Appl. Algebra 216(3) (2012),718–733.[10]
D. S¸tefan , Hochschild cohomology on Hopf Galois extensions , J. Pure Appl. Alge-bra, 103 (1995), pp. 221–233.[11]
R. Taillefer , Injective Hopf bimodules, Cohomologies of Infinite DimensionalHopf Algebras and Graded-commutativity of the Yoneda Product , Journal of Algebra.276 (2004), no.1, 259-279.[12]
Y. Volkov , Gerstenhaber bracket on the Hochschild cohomology via an arbitraryresolution , Proc. Edinburgh Math. Soc. (2) 62 (3) (2019), 817-836.[13]
Y. Volkov, S. Witherspoon , Graded Lie structure on cohomology of some exactmonoidal categories , arXiv:2004.06225.[14]
S. Witherspoon , Hochschild Cohomology for Algebras , volume 204. AmericanMathematical Society, 2019.
Department of Mathematics, Texas A&M University, College Station, Texas 77843,USA
E-mail address : [email protected]@math.tamu.edu