BV-differential on Hochschild cohomology of Frobenius algebras
aa r X i v : . [ m a t h . K T ] M a y BV-differential on Hochschild cohomology of Frobeniusalgebras.
Y. V. Volkov ∗ Abstract
For a finite-dimensional Frobenius k -algebra R with the Nakayama automorphism ν we define an algebra HH ∗ ( R ) ν ↑ . If the order of ν is not divisible by the characteristicof k , this algebra is isomorphic to the Hochschild cohomology algebra of R .. We provethat this algebra is a BV-algebra. We use this fact to calculate the Gerstenhaberalgebra structure and BV-structure on the Hochschild cohomology algebras of a familyof self-injective algebras of tree type D n . Hochschild cohomology is a subtle invariant of an associative algebra which carries a lot ofinformation about its structure. The cohomology theory of associative algebras was intro-duced by Hochschild. It was later shown in [1], that the Hochschild cohomology algebra isa Gerstenhaber algebra. Sometimes we can define a BV-differential on the Hochschild coho-mology algebra in such way that the structure of the Gerstenhaber algebra can be definedusing this differential. In [2] Tradler shows that such differential can be defined for symmet-ric algebras. It is proved in [3] that a BV-differential can be constructed if the given algebrais a Frobenius algebra with finite stable Calabi-Yau dimension and periodic Hochschild co-homology. Moreover it is shown in [4] that a BV-differential exists in the case of Calabi-Yaualgebra.Let R be a Frobenius algebra over an algebraically closed field. Let ν be its Nakayamaautomorphism. In this paper we define an algebra HH ∗ ( R ) ν ↑ . It turns out that HH ∗ ( R ) ν ↑ ≃ HH ∗ ( R ) in many cases. The Lie bracket on the algebra HH ∗ ( R ) induces a bracket onHH ∗ ( R ) ν ↑ . We slightly modify the Tradler’s proof for symmetric algebras to construct theBV-differential on the algebra HH ∗ ( R ) ν ↑ .In section 5 we apply the obtained results to describe the Gerstenhaber algebra structureand the BV-differential for a family of self-injective algebras of tree type D n . The Hochschildcohomology ring for these algebras was described in terms of generators and relations in[5]. Note that the calculation of these structures is a difficult task. There are only a fewexamples of such calculations. The structure of Gerstenhaber algebra was described for serialself-injective algebras in [6]. The calculus structure which includes the Gerstenhaber algebrastructure was calculated for preprojective algebras of type A n , D n , E n and L n in [7] and [8].Furtermore the Gerstenhaber algebra structure and the BV-differential for the group algebraof the quaternion group of order 8 over a field of characteristic 2 were calculated in [9]. ∗ The author was supported by RFBR (13-01-00902 A and 14-01-31084 mol a). Basic definitions and constructions
Throughout the paper we suppose that k is an algebraically closed field, R is a finite-dimensional k -algebra, Λ = R ⊗ R op is the enveloping algebra of R (we write ⊗ instead of ⊗ k ). The bar-resolutionBar ∗ ( R ) : ( R µ ← ) R ⊗ d ← R ⊗ ← · · · ← R ⊗ ( n +2) d n ← R ⊗ ( n +3) ← · · · of the algebra R is defined in the following way: Bar n ( R ) = R ⊗ ( n +2) ( n > µ : R ⊗ R → R is the multiplication of R and d n ( n >
0) is defined by the formula d n ( a ⊗ · · · ⊗ a n +2 ) = n +1 X i =0 ( − i a ⊗ · · · ⊗ a i − ⊗ a i a i +1 ⊗ a i +2 ⊗ · · · ⊗ a n +2 , where a i ∈ R (0 i n + 2). The homology of the complex C ∗ ( R ) = Hom Λ (Bar ∗ ( R ) , R )is called Hochschild cohomology of the algebra R . Note that C ( R ) ≃ R and C n ( R ) =Hom Λ ( R ⊗ ( n +2) , R ) ≃ Hom k ( R ⊗ n , R ). Let introduce notation δ in ( f )( a ⊗ · · · ⊗ a n +1 ):= a f ( a ⊗ · · · ⊗ a n +1 ) , if i = 0 , ( − i f ( a ⊗ · · · ⊗ a i a i +1 ⊗ · · · ⊗ a n +1 ) , if 1 i n, ( − n +1 f ( a ⊗ · · · ⊗ a n ) a n +1 , if i = n + 1 . for f ∈ C n ( R ). Then C ∗ ( R ) has the form R δ → Hom k ( R, R ) → · · · → Hom k ( R ⊗ n , R ) δ n → Hom k ( R ⊗ ( n +1) , R ) → · · · , where δ n = n +1 P i =0 δ in .The cup product f ⌣ g ∈ C n + m ( R ) = Hom k ( R ⊗ ( n + m ) , R ) for f ∈ C n ( R ) and g ∈ C m ( R )is given by ( f ⌣ g )( a ⊗ · · · ⊗ a n + m ) := f ( a ⊗ · · · ⊗ a n ) · g ( a n +1 ⊗ · · · ⊗ a n + m ) . This cup product induces a well-defined product on Hochschild cohomology ⌣ : HH n ( R ) × HH m ( R ) −→ HH n + m ( R )which turns the graded k -vector space HH ∗ ( R ) = L n ≥ HH n ( R ) into a graded commutativealgebra ([1, Corollary 1]).Let now define the Lie bracket. Let f ∈ C n ( R ), g ∈ C m ( R ). If n, m ≥
1, then for1 ≤ i ≤ n , we set( f ◦ i g )( a ⊗ · · · ⊗ a n + m − ):= f ( a ⊗ · · · ⊗ a i − ⊗ g ( a i ⊗ · · · ⊗ a i + m − ) ⊗ a i + m ⊗ · · · ⊗ a n + m − );if n ≥ m = 0, then g ∈ R and for 1 ≤ i ≤ n , we set( f ◦ i g )( a ⊗ · · · ⊗ a n − ) := f ( a ⊗ · · · ⊗ a i − ⊗ g ⊗ a i ⊗ · · · ⊗ a n − ) . f ◦ g := n X i =1 ( − ( m − i − f ◦ i g and [ f, g ] := f ◦ g − ( − ( n − m − g ◦ f. Note that [ f, g ] ∈ C n + m − ( R ). Then [ , ] induces a well-defined Lie bracket on Hochschildcohomology [ , ] : HH n ( R ) × HH m ( R ) −→ HH n + m − ( R )such that (HH ∗ ( R ) , ⌣, [ , ]) is a Gerstenhaber algebra ([1]).A Batalin–Vilkovisky algebra (BV-algebra for short) is a Gerstenhaber algebra ( A • , ⌣, [ , ]) together with an operator ∆ : A • → A •− of degree − ◦ ∆ = 0 and[ a, b ] = − ( − ( | a |− | b | (∆( a ⌣ b ) − ∆( a ) ⌣ b − ( − | a | a ⌣ ∆( b )) (2.1)for homogeneous elements a, b ∈ A • . The following Theorem is proved in [2]. Theorem 1. [2, Theorem 1] Let R be a symmetric algebra, i.e. an algebra with a nondegener-ate associative symmetric bilinear form h , i : R × R → k . For f ∈ C n ( R ) = Hom k ( R ⊗ n , R ) define ∆( f ) ∈ C n − ( R ) = Hom k ( R ⊗ ( n − , R ) by the formula h ∆( f )( a ⊗ · · · ⊗ a n − ) , a n i = n X i =1 ( − i ( n − h f ( a i ⊗ · · · ⊗ a n ⊗ a ⊗ · · · ⊗ a i − ) , i , where a i ∈ R ( i n ). The map ∆ induces the differential ∆ : HH n ( R ) → HH n − ( R ) .Then (HH ∗ ( R ) , ⌣, [ , ] , ∆) is a BV-algebra. If σ : R → R is an automorphism of the algebra R , then we can define a map φ σ : C n ( R ) → C n ( R ) by the formula (cid:0) φ σ ( f ) (cid:1) ( a ⊗ · · · ⊗ a n ) = σ − ( f ( σ ( a ) ⊗ · · · ⊗ σ ( a n ))) , where f ∈ C n ( R ) = Hom k ( R ⊗ n , R ), a i ∈ R (1 i n ). From here on we write f σ insteadof φ σ ( f ). It is easy to show that ( δ n f ) σ = δ n f σ for f ∈ C n ( R ). Consequently, the map φ induces a map on Hochschild cohomology ( ) σ : HH n ( R ) → HH n ( R ). For a map σ : X → X we denote by X σ the set { x ∈ X | σ ( x ) = x } . Then δ n : C n ( R ) → C n +1 ( R ) induces themap δ σn : C n ( R ) σ → C n +1 ( R ) σ . We denote by HH n ( R ) σ ↑ the homology of the complex( C n ( R ) σ , δ σn ) and defineHH ∗ ( R ) σ ↑ := M n > HH n ( R ) σ ↑ = M n > Ker( δ σn ) / Im ( δ σn − ) . It is proved in [10] that ⌣ : C n ( R ) × C m ( R ) −→ C n + m ( R ) defined above determines analgebra structure on HH ∗ ( R ) σ ↑ and the inclusion of C n ( R ) σ into C n ( R ) induces an algebrahomomorphism Θ σR : HH ∗ ( R ) σ ↑ → HH ∗ ( R ) σ . σR is bijective if ord( σ ) < ∞ and char k ord( σ ). The fact that [ , ] : C n ( R ) × C m ( R ) −→ C n + m − ( R ) induces a Gerstenhaber algebra structure on HH ∗ ( R ) σ ↑ can be provedanalogously to the fact that [ , ] induces a Gerstenhaber algebra structure on HH ∗ ( R ) (cf.[1]). It is easy to see that Θ σR is a homomorphism of Gerstenhaber algebras.If d : M → N is a morphism of R -bimodules and σ , σ are automorphisms of thealgebra R , then we denote by d ( σ ,σ ) : σ M σ → σ N σ the morphism of R -bimodules, whichis determined by the formula d ( σ ,σ ) ( m ) = d ( m ) for m ∈ M (we denote by σ M σ thebimodule which is equal to M as k -linear space with multiplication ∗ defined by the equality a ∗ m ∗ b = σ ( a ) mσ ( b ) for m ∈ M , a, b ∈ R ).Algebra R is called a Frobenius algebra if there is a linear map ǫ : R → k such that thebilinear form h a, b i = ǫ ( ab ) is nondegenerated. The Nakayama automorphism ν : R → R isthe automorphism which satisfies the equation h a, b i = h b, ν ( a ) i for all a, b ∈ R . From hereon we assume that R is a Frobenius algebra, h , i is the corresponding bilinear form and ν isthe Nakayama automorphism defined by it. Let f ∈ C n ( R ), n >
1. Define ∆ i f ∈ C n − ( R ) by the equation h ∆ i f ( a ⊗ · · · ⊗ a n − ) , a n i = h f ( a i ⊗ · · · ⊗ a n − ⊗ a n ⊗ νa ⊗ · · · ⊗ νa i − ) , i , where a i ∈ R (1 i n ). Further define∆ := n X i =1 ( − i ( n − ∆ i : C n ( R ) → C n − ( R ) . (3.1) Lemma 1. δ n − (∆ f ) + ∆ δ n ( f ) = f ν − f ∀ f ∈ C n ( R ) .Proof. Let f ∈ C n ( R ), a i ∈ R (1 i n + 1). Set A i := a i ⊗ · · · ⊗ a n +1 ⊗ νa ⊗ · · · ⊗ νa i − ∈ R ⊗ ( n +1) . Direct calculations show that h ( − i ( n − δ jn − (∆ i f )( a ⊗ · · · ⊗ a n ) , a n +1 i = ( h ( − ( i +1) n +1 δ n − i + j +1 n ( f )( A i +1 ) , i , if 0 j i − h ( − in +1 δ j − i +1 n ( f )( A i ) , i , if i j n .Adding these equalities for 0 j n and 1 i n , we obtain h δ n − (∆ f )( a ⊗ · · · ⊗ a n ) , a n +1 i + h ∆ δ n ( f )( a ⊗ · · · ⊗ a n ) , a n +1 i = * n +1 X i =1 ( − in +1 n X j =1 δ jn ( f )( A i ) , + + * n +1 X i =1 ( − in δ n ( f )( A i ) , + = * n +1 X i =1 ( − in ( δ n ( f )( A i ) + δ n +1 n ( f )( A i )) , + . h δ n ( f )( A i ) + ( − n δ n +1 n ( f )( A i +1 ) , i = 0 for 1 i n , we have h δ n − (∆ f )( a ⊗ · · · ⊗ a n ) , a n +1 i + h ∆ δ n ( f )( a ⊗ · · · ⊗ a n ) , a n +1 i = h ( − n δ n +1 n ( f )( A ) + δ n ( f )( A n +1 ) , i = h a n +1 f ( νa ⊗ · · · ⊗ νa n ) , i − h f ( a ⊗ · · · ⊗ a n ) a n +1 , i = h ( f ν − f )( a ⊗ · · · ⊗ a n ) , a n +1 i . The next two corollaries follow directly from Lemma 1.
Corollary 1.
The map ∆ defined by the formula (3.1) induces a map ∆ : HH n ( R ) ν ↑ → HH n − ( R ) ν ↑ . Corollary 2. HH ∗ ( R ) ν = HH ∗ ( R ) . For a special case, the second corollary is proved in [11]. Now we are able to formulatethe following generalization of Theorem 1.
Theorem 2.
Let R be a Frobenius algebra with bilinear form h , i and Nakayama auto-morphism ν . Then ∆ defined by (3.1) induces a BV -algebra structure on the Gerstenhsberalgebra (HH ∗ ( R ) ν ↑ , ⌣, [ , ]) .Proof. Let us prove that ∆ ◦ ∆ = 0 in HH ∗ ( R ) ν ↑ . It is enough to prove that any elementof HH n ( R ) ν ↑ can be represented by an element f ∈ Ker δ νn such that f ( a ⊗ · · · ⊗ a n ) = 0 if a i = 1 for some 1 i n . Let us define the maps s in : C n +1 ( R ) → C n ( R ) (0 i n ) bythe formula ( s in ( g ))( a ⊗ · · · ⊗ a n ) = ( − i g ( a ⊗ · · · ⊗ a i ⊗ ⊗ a i +1 · · · ⊗ a n )for g ∈ C n +1 ( R ). It is clear that s in induces a map s i,νn : C n +1 ( R ) ν → C n ( R ) ν . Now we canprove by induction on N that any element f ′ ∈ HH n ( R ) ν ↑ can be represented by an element f ∈ Ker δ νn such that f ( a ⊗ · · · ⊗ a n ) = 0 if a i = 1 for some 1 i N . Indeed, if N = 0,then the required assertion is obvious. Now assume that the cohomology class of f is equalto f ′ and f ( a ⊗ · · · ⊗ a n ) = 0 if a i = 1 for some 1 i N −
1. Then direct calculationsshow that ( f − δ νn − s N − ,νn − ( f ))( a ⊗ · · · ⊗ a n ) = ( s N − ,νn δ νn ( f ))( a ⊗ · · · ⊗ a n ) = 0if a i = 1 for some 1 i N .Now assume that f ∈ C n ( R ) ν , g ∈ C m ( R ) ν . Let us introduce the following notation∆ ′ ( f ⊗ g ) := m X i =1 ( − i ( n + m − ∆ i ( f ⌣ g ) . Then ∆( f ⌣ g ) = ∆ ′ ( f ⊗ g ) + ( − nm ∆ ′ ( g ⊗ f ).The remaining part of the proof is analogous to the proof of Theorem 1 (cf. [2]).Note that a Frobenius algebra is symmetric if and only if its Nakayama automorphismequals Id R for some bilinear form. So Theorem 1 is a special case of Theorem 2. Corollary 3. If ord( ν ) < ∞ and char k ord( ν ) , then the map defined by the formula (3.1) induces a BV-differential on the Gerstenhaber algebra (HH ∗ ( R ) , ⌣, [ , ]) .Proof. This follows from Corollary 2 of Lemma 1 and the fact that Θ νR : HH ∗ ( R ) ν ↑ → HH ∗ ( R ) ν is an isomorphism of Gerstenhaber algebras if ord( ν ) < ∞ and char k ord( ν ).5 On a family of self-injective algebras of tree type D n Let n > r >
1. Let us define a k -algebra R = R ( n, r ). Consider a quiver with relations( Q , I ). Its set of vertices is Q = Z r × { j ∈ N | j n } . The set of arrows Q of thequiver Q consists of the following elements: γ i,p : ( i, n − → ( i, p ) , β i,p : ( i, p ) → ( i + 1 , ,α i,j : ( i, j ) → ( i, j + 1) (1 i r, p ∈ { n − , n } , j n − . In addition we use the following auxiliary notation: τ i = β i,n γ i,n , ω i,j ,j = α i,j . . . α i,j , µ i,j = ω i,j, , η i,j = ω i,n − ,j . From here on for the uniformity of the notation we additionally suppose that the emptyproduct of the arrows of the quiver is identified with an appropriate idempotent of thealgebra R ; for example, µ i, = e i, , ω i,j − ,j = e i,j , η i,n − = e i,n − . Denote by φ : { , . . . , n } →{ , . . . , n } the map such that φ ( j ) = j for 1 j n − φ ( n −
1) = n and φ ( n ) = n − I is generated by the elements γ i,φ ( p ) η i, β i − ,p , β i,n − γ i,n − − τ i , µ i +1 ,j τ i η i,j (1 i r, j n − , p ∈ { n − , n } ) . i, n − β i,n − y y sssss i+1,2 α i +1 , u u ❦❦❦❦❦❦❦❦❦❦❦❦ i+1,1 α i +1 , o o i,n-2 γ i,n − e e ❑❑❑❑❑ γ i,n y y ssssss i,n-3 α i,n − o o i, n β i,n e e ❑❑❑❑❑❑ α i,n − h h ❘❘❘❘❘❘❘❘❘❘❘ α r,n − ❅❅❅❅❅❅ r, n − β r,n − % % ❑❑❑❑❑❑ , n − β ,n − $ $ ■■■■■ r,n-2 γ r,n − tttttt γ r,n % % ❏❏❏❏❏❏❏ α , / / · · · α ,n − / / γ ,n − ssssss γ ,n % % ❑❑❑❑❑❑ α , @ @ ✂✂✂✂✂✂ r, n β r,n sssssss , n β ,n : : ✉✉✉✉✉✉✉ Set R = R ( n, r ) = k Q /I . It is easily verified that R is a Frobenius algebra. For two paths w and w of the quiver Q the bilinear form h , i is defined in the following way: h w , w i = ǫ ( w w ) := ( , if w w is a nonzero path of length n − ν . This automorphism is definedon idempotents and arrows by the formulas ν ( e i,t ) = e i − ,t , ν ( α i,j ) = α i − ,j , ν ( γ i,p ) = γ i − ,p , ν ( β i,p ) = β i − ,p (1 i r, t n, j n − , p ∈ { n − , n } ) . It is clear that the set B R = { µ i +1 ,t − τ i η i,j | i r, t j n − }∪ { ω i,t − ,j | i r, j t n − }∪ { γ i,p η i,j | i r, j n − , p ∈ { n − , n }}∪ { µ i +1 ,j − β i,p | i r, j n − , p ∈ { n − , n }}∪ { γ i,p µ i,n − γ i − ,p | i r, p ∈ { n − , n }}
6s a k -basis of R . We define for b ∈ B R the element ¯ b ∈ B R by the equalities µ i +1 ,t − τ i η i,j = ω i +1 ,j − ,t (1 i r, t j n − ω i,t − ,j = µ i +1 ,j − τ i η i,t (1 i r, j t n − γ i,p η i,j = µ i +1 ,j − β i,p (1 i r, j n − , p ∈ { n − , n } ) µ i +1 ,j − β i,n − = γ i +1 ,p η i +1 ,j (1 i r, j n − , p ∈ { n − , n } ) γ i,p µ i,n − γ i − ,p = e i,p (1 i r, p ∈ { n − , n } ) . It is easy to show that h a, b i = ( , if b = ¯ a ,0 , if b = ¯ a , for a, b ∈ B R .The Hochschild cohomology algebra of R ( n, r ) is described in [12] and [5]. Let us recallsome results of these works. Denote by e x the idempotent of the algebra R corresponding toa vertex x of the quiver Q . Then { e x ⊗ e y } x,y is a full set of orthogonal primitive idempotentsfor the algebra Λ. Denote by P [ x ][ y ] = Λ e x ⊗ e y the projective Λ-module, which correspondsto idempotent e x ⊗ e y . Let σ : R → R be the automorphism of R , which is defined on theidempotents and arrows by the formulas σ ( e i,j ) = e i + n − ,φ n ( j ) , σ ( α i,j ) = α i + n − ,j , σ ( γ i,p ) = − γ i + n − ,φ n ( p ) ,σ ( β i,p ) = β i + n − ,φ n ( p ) (1 i r, j n − , p ∈ { n − , n } ) . Denote by Q t the t -th module in the minimal projective bimodule resolution of R . Then Q m = r M i =1 (cid:18)(cid:16) n − − m M j =1 P [ i + m,j + m ][ i,j ] (cid:17) ⊕ (cid:16) n − M j = n − − m P [ i + m,j + m − ( n − i,j ] (cid:17) ⊕ P [ i + m,φ m ( n − i,n − ⊕ P [ i + m,φ m ( n )][ i,n ] (cid:19) (0 m n − ,Q m +1 = r M i =1 (cid:18)(cid:16) n − − m M j =1 P [ i + m,j + m +1][ i,j ] (cid:17) ⊕ P [ i + m,n − i,n − − m ] ⊕ P [ i + m,n ][ i,n − − m ] ⊕ (cid:16) n − M j = n − − m P [ i + m +1 ,j + m − ( n − i,j ] (cid:17) ⊕ P [ i + m +1 ,m +1][ i,n − ⊕ P [ i + m +1 ,m +1][ i,n ] (cid:19) (0 m n − . Moreover, Q t + l (2 n − = σ l ( Q t ) for 0 t n − l >
0. The definitions of the differentials d Qt : Q t +1 → Q t can be found in [12]. The augmentation map µ : Q → R is definedby the formula µ ( a ⊗ b ) = ab ∈ R for a ⊗ b ∈ P [ x ][ x ] ( x ∈ Q ). It is easy to check that ν ( P [ i ,j ][ i ,j ] ) ν ≃ P [ i +1 ,j ][ i +1 ,j ] . These isomorphisms give isomorphisms θ t : Q t → ν ( Q t ) ν for t >
0. Using the description of the differentials d Qt it is easy to verify that θ ∗ is a chainmap, which lifts the isomorphism φ ν : R → ν R ν defined by the formula φ ν ( a ) = νa for a ∈ R . Let f ∈ Hom Λ ( Q t , R ), f d Qt = 0 and f is defined on the direct summands of Q t by the7qualities f ( e [ i ,j ][ i ,j ] ) = f [ i ,j ][ i ,j ] ∈ R . It follows from [10, Lemma 2] that f ν is definedby the equalities f ν ( e [ i ,j ][ i ,j ] ) = ν − f [ i − ,j ][ i − ,j ] . To calculate the Lie bracket we will partially use the algorithm described in [9]. Toapply this algorithm we need the homomorphism of left R -modules D t : Q t → Q t +1 ( t > D − : R → Q , which satisfy the equations D t +1 D t = 0 ( t > , D t − d Qt − + d Qt D t = Id Q t ( t > ,D − µ + d Q D = Id Q , µD − = Id R . (4.1)Let D − : R → Q be a homomorphism of left modules such that D − ( e x ) = e [ x ][ x ] for all x ∈ Q .Now let us define the homomorphisms D t ( t > e x ⊗ b ( x ∈ Q , b ∈ B R ). Let 0 m n −
3. Then D m ( e i + m,j + m ⊗ e i,j b ) (1 i r, j n − − m ) is equal to j − X s = q ω i + m,j + m − ,s + m +1 ⊗ ω i,s − ,q if b = ω i,j − ,q , 1 q j , j − X s =1 ω i + m,j + m − ,s + m +1 ⊗ µ i,s − β i − ,p + ω i + m,j + m − ,m +1 ⊗ e i − ,p if b = µ i,j − β i − ,p , p ∈ { n − , n } , j − X s =1 ω i + m,j + m − ,s + m +1 ⊗ µ i,s − τ i − η i − ,q + ω i + m,j + m − ,m +1 ⊗ γ i − ,n − η i − ,q − ω i + m,j + m − ,q + m − ( n − ⊗ e i − ,q if b = µ i,j − τ i − η i − ,q , n − − m q n − j − X s =1 ω i + m,j + m − ,s + m +1 ⊗ µ i,s − τ i − η i − ,q + ω i + m,j + m − ,m +1 ⊗ γ i − ,n − η i − ,q + µ i + m,j + m − β i + m − ,φ m ( n − ⊗ ω i − ,n − − m,q + n − − m X s = q µ i + m,j + m − τ i + m − η i + m − ,s + m +1 ⊗ ω i − ,s − ,q if b = µ i,j − τ i − η i − ,q , j q n − − m ; D m ( e i + m,j + m − ( n − ⊗ e i,j b ) (1 i r, n − − m j n −
2) is equal to 0 if b = µ i,j − τ i − η i − ,j , and is equal to − e i + m,j + m − ( n − ⊗ e i − ,j if b = µ i,j − τ i − η i − ,j ; D m ( e i + m,φ m ( p ) ⊗ e i,p b ) (1 i r, p ∈ { n − , n } ) is equal to 08f b = e i,p or b = γ i,p η i,q , n − − m q n − e i + m,φ m ( p ) ⊗ ω i,n − − m,q + n − − m X s = q γ i + m,φ m ( p ) η i + m,s + m +1 ⊗ ω i,s − ,q if b = γ i,p η i,q , 1 q n − − m , e i + m,φ m ( p ) ⊗ µ i,n − − m β i − ,p + n − − m X s = q γ i + m,φ m ( p ) η i + m,s + m +1 ⊗ µ i,s − ,q β i − ,p + γ i + m,φ m ( p ) η i + m,m +1 ⊗ e i − ,p if b = γ i,p η i, β i − ,p .Let 0 m n −
3. Then D m +1 ( e i + m,j + m +1 ⊗ e i,j b ) (1 i r, j n − − m ) is equal to 0 if b = µ i,j − τ i − η i − ,j ,and is equal to e i + m,j + m +1 ⊗ e i − ,j if b = µ i,j − τ i − η i − ,j ; D m +1 ( e i + m +1 ,j + m − ( n − ⊗ e i,j b ) (1 i r, n − − m j n −
2) is equal to j − X s = q ω i + m +1 ,j + m − ( n − ,s + m +( n − ⊗ ω i,s − ,q if b = ω i,j − ,q , n − − m q j , j − X s = n − − m ω i + m +1 ,j + m − ( n − ,s + m +( n − ⊗ ω i,s − ,q if b = ω i,j − ,q , 1 q n − − m , j − X s = n − − m ω i + m +1 ,j + m − ( n − ,s + m +( n − ⊗ µ i,s − β i − ,n − + µ i + m +1 ,j + m − ( n − β i + m,φ m +1 ( n − ⊗ e i − ,n − if b = µ i,j − β i − ,n − , j − X s = n − − m ω i + m +1 ,j + m − ( n − ,s + m +( n − ⊗ µ i,s − β i − ,n − µ i + m +1 ,j + m − ( n − β i + m,φ m +1 ( n ) ⊗ e i − ,n if b = µ i,j − β i − ,n , j − X s = n − − m ω i + m +1 ,j + m − ( n − ,s + m +( n − ⊗ µ i,s − τ i − η i − ,q − µ i + m +1 ,j + m − ( n − β i + m,φ m +1 ( n ) ⊗ γ i − ,n η i − ,q + µ i + m +1 ,j + m − ( n − β i + m,φ m +1 ( n − ⊗ γ i − ,n − η i − ,q − n − X s = q µ i + m +1 ,j + m − ( n − τ i + m η i + m,s + m − ( n − ⊗ ω i − ,s − ,q b = µ i,j − τ i − η i,q ( j q n − D m +1 ( e i + m +1 ,m +1 ⊗ e i,n − b ) (1 i r ) is equal to 0if b = γ i,n − η i, β i − ,n − , and is equal to n − X s = n − − m ω i + m +1 ,m,s + m − ( n − ⊗ µ i,s − β i − ,n − + µ i + m +1 ,m β i + m,φ m +1 ( n − ⊗ e i − ,n − if b = γ i,n − η i, β i − ,n − ; D m +1 ( e i + m +1 ,m +1 ⊗ e i,n b ) (1 i r ) is equal to 0if b = e i,n , − n − X s = q ω i + m +1 ,m,s + m +( n − ⊗ ω i,s − ,q if b = γ i,n η i,q , n − − m q n − − n − X s = n − − m ω i + m +1 ,m,s + m +( n − ⊗ ω i,s − ,q if b = γ i,n η i,q , 1 q n − − m , − n − X s = n − − m ω i + m +1 ,m,s + m − ( n − ⊗ µ i,s − β i − ,n − + µ i + m +1 ,m β i + m,φ m +1 ( n ) ⊗ e i − ,n if b = γ i,n η i, β i − ,n ; D m +1 ( e i + m,φ m ( p ) ⊗ e i,n − − m b ) (1 i r, p ∈ { n − , n } ) is equal to 0if b = ω i,n − − m,q , 1 q n − − m or b = µ i,n − − m β i − ,p , e i + mφ m ( p ) ⊗ e i − ,φ ( p ) if b = µ i,n − − m β i − ,φ ( p ) , e i + mφ m ( n ) ⊗ γ i − ,n − η i,q if p = n , b = µ i,n − − m τ i − η i − ,q ( n − − m q n − e i + mφ m ( n − ⊗ γ i − ,n η i,q + n − X s = q γ i + mφ m ( n − η i + m,s + m − ( n − ⊗ ω i − ,s − ,q if p = n − b = µ i,n − − m τ i − η i − ,q ( n − − m q n − D n − in the following way. D n − ( e i + n − ,j ⊗ e i,j b ) is equal to 0 if b = µ i,j − τ i − η i,j , and is equal to − e i + n − ,j ⊗ e i − ,j if b = µ i,j − τ i − η i,j ; D n − ( e i + n − ,φ n − ( n − ⊗ e i,n − b ) is equal to 0 if b = γ i,n − η i − , β i − ,n − , and is equal to − e i + n − ,φ n − ( n − ⊗ e i − ,n − if b = γ i,n − η i − , β i − ,n − ;10 n − ( e i + n − ,φ n − ( n ) ⊗ e i,n b ) is equal to 0 if b = γ i,n η i − , β i − ,n , and is equal to e i + n − ,φ n − ( n ) ⊗ e i − ,n if b = γ i,n η i − , β i − ,n .Moreover, D t + l (2 n − = ( D t ) ( σ l , for 0 t n − l >
0. Direct calculations show thatthe maps D t ( t > −
1) satisfy the equalities (4.1).Let θ Bar ,t : Bar n ( R ) → ν (Bar n ( R )) ν be defined by the formula θ Bar ,t ( a ⊗ · · · ⊗ a t +1 ) = νa ⊗ · · · ⊗ νa t +1 . It is easy to show that the chain maps Φ ∗ : Q ∗ → Bar ∗ ( R ) and Ψ ∗ :Bar ∗ ( R ) → Q ∗ constructed using the maps d Qt and D t ( t >
0) by the algorithm from [9]satisfy the equalities θ Bar ,t Φ t = (Φ t ) ( ν,ν ) θ t and θ t Ψ t = (Ψ t ) ( ν,ν ) θ Bar ,t . Then it follows from [10,Lemma 2] that t -cocycle lies in Im Θ νR if and only if it can be represented by a homomorphism f ∈ Hom Λ ( Q t , R ) such that f ( e [ i ,j ][ i ,j ] ) = ν − f [ i − ,j ][ i − ,j ] . (4.2) In this section we suppose that R = R ( n, r ). Let w be a path from a vertex x to a vertex y . Denote by w ∗ the element of Hom Λ ( P [ y ][ x ] , R ) such that w ∗ ( e y ⊗ e x ) = w . For 1 i r consider the following auxiliary homomorphisms: w i,m,j = ( ω i,j + m − ,j ) ∗ (0 m n − , j n − − m ); t i,m,j = ( µ i +1 ,j + m − ( n − τ i η i,j ) ∗ (1 m n − , n − − m j n − u i,m,q = ( γ i,q η i,n − − m ) ∗ (0 m n − , q ∈ { n − , n } ); v i,m,q = ( µ i +1 ,m β i,q ) ∗ (0 m n − , q ∈ { n − , n } ); u i,q = e i,q ∗ ( q ∈ { n − , n } ); v i,q = ( γ i +1 ,q η i +1 , β i,q ) ∗ ( q ∈ { n − , n } ) . Let now define some elements of the algebra HH ∗ ( R ).a) Define 1-cocycle ε ∈ Hom Λ ( Q , R ) by the formula ε = u , ,n − + u , ,n . b) Let s = 2 m + l (2 n − m n − r | m + l ( n − | m + ln and one of thefollowing conditions is satisfied: char k = 2 or 2 | l . Define s -cocycle f s ∈ Hom Λ ( Q s , R ) bythe formula f s = r X i =1 (cid:16) n − − m X j =1 w i,m,j + u i,n − + u i,n (cid:17) . c) Let s = 2 m + 1 + l (2 n − m n − r | m + l ( n − m + ln and one of thefollowing conditions is satisfied: char k = 2 or 2 l . Define s -cocycle g s ∈ Hom Λ ( Q s , R ) bythe formula g s = r X i =1 (cid:16) n − X j = n − − m t i,m,j + u i,m,n − + v i,m,n − (cid:17) .
11) Let s = ( l + 1)(2 n − − r | ( l + 1)( n − −
1, 2 ( l + 1) n . Define s -cocycle h s ∈ Hom Λ ( Q s , R ) by the formula h s = r X i =1 n − X j =1 ( − j w i, ,j . e) Let s = ( l + 1)(2 n − − r | ( l + 1)( n − −
1, 2 | n and one of the following conditionsis satisfied: char k = 2 or 2 | l . Define s -cocycle p s ∈ Hom Λ ( Q s , R ) by the formula p s = r X i =1 (cid:16) n − X j =1 ( − j w i, ,j + u i,n − (cid:17) . f) Let s = l (2 n − l > r | l ( n − −
1, 2 | ln and either char k = 2 or 2 l . Denote by χ s ∈ Hom Λ ( Q s , R ) the s -cocycle, which is equal to v r,n on P [1 ,n ][ r,n ] and is equal to 0 on otherdirect summands of Q s .g) Let s = l (2 n − r | l ( n − −
1, 2 ln . Denote by ξ s ∈ Hom Λ ( Q s , R ) the s -cocycle,which is equal to t r,n − , on P [1 , r, and is equal to 0 on other direct summands of Q s .h) Let r = 1. For 1 j n − ε ( j )0 the 0-cocycle, which is equal to t ,n − ,j on P [1 ,j ][1 ,j ] and is equal to 0 on other direct summands of Q . For q ∈ { n − , n } denote by ε ( q )0 the 0-cocycle, which is equal to v ,q on P [1 ,q ][1 ,q ] and is equal to 0 on other direct summandsof Q .It was shown in [5] that the elements defined in a)–h) are cocycles for the correspondingvalues of s and that they generate HH ∗ ( R ) as a k -algebra. In addition, ξ s can be excludedfrom the set of generators in the case where char k n − and χ s can be excluded from the setof generators in the case where 2 | n , char k n −
1. Moreover, it is proved in the same workthat the elements of the form f s , g s , h s , p s , ε f s , ε g s , χ s , ξ s and ε ( q )0 generate HH ∗ ( R ) as a k -linear space.If char k r , then HH ∗ ( R ) is a BV-algebra by the Corollary of Theorem 2. Since f s , g s , h s and p s satisfy the condition (4.2), they lie in the image of Θ νR (even if char k | r ).Let us introduce the following notation F ( x ) = , if x = ε , m + l ( n − , if x = f m + l (2 n − or x = g m +1+ l (2 n − , l ( n − − , if x = p l (2 n − − , x = h l (2 n − − ,x = ξ l (2 n − or x = χ l (2 n − .Note that r | F ( x ) in all the cases. Proposition 1.
Let x ∈ { ε , f s , g s , h s , p s } . Then [ x, ε ] = F ( x ) r x (5.1) in HH ∗ ( R ) . roof. Recall the construction of the chain maps Φ t : Q t → Bar t ( R ) and Ψ t : Bar t ( R ) → Q t from [9].Firstly, define Φ and Ψ by the equalities Φ ( e x ⊗ e x ) = e x ⊗ e x , Ψ ( e x ⊗ e x ) = e x ⊗ e x ,Ψ ( e x ⊗ e y ) = 0 for x, y ∈ Q , x = y . For t > t is defined by the formulaΨ t (1 ⊗ a ⊗ · · · ⊗ a n ⊗
1) = D t − (Ψ t − (1 ⊗ a ⊗ · · · ⊗ a n − ⊗ a n ) . (5.2)From now on we assume that Q t = L p ∈ X t P [ x p ][ y p ] for all t >
0, where { X t } t > is a set of disjointsets. Let π p : Q t → P [ x p ][ y p ] be the canonical projection and π p ′ d Qt − ( e x p ⊗ e y p ) = a p,p ′ ⊗ e y p ′ + b p,p ′ ⊗ c p,p ′ for p ′ ∈ X t − , p ∈ X t , where a p,p ′ , b p,p ′ ∈ R and c p,p ′ are in Jacobson radical of R . ThenΦ t ( e x p ⊗ e y p ) = X p ′ ∈ X t − b p,p ′ Φ t − ( e x p ′ ⊗ e y p ′ ) c p,p ′ ⊗ . (5.3)Consequently, Ψ t Φ t ( e x p ⊗ e y p ) = X p ′ ∈ X t − b p,p ′ D t − (Ψ t − Φ t − ( e x p ′ ⊗ e y p ′ ) c p,p ′ ) . It follows from these formulas and induction on t that for all t > p ∈ X t we haveΦ t ( e x p ⊗ e y p ) = e x p ⊗ a p, ⊗ · · · ⊗ a p,t ⊗ e y p + X z ∈ Y p a z, ⊗ · · · ⊗ a z,t +1 , (5.4)where Ψ t ( e x p ⊗ a p, ⊗ · · · ⊗ a p,t ⊗ e y p ) = e x p ⊗ e y p and Ψ t ( a z, ⊗ · · · ⊗ a z,t +1 ) = 0 for all z ∈ Y p .The equality [ ε , ε ] = 0 follows from the definition of the Lie bracket and the fact that ε is an element of odd degree. For other elements we use the formula[ f, ε ] = (( f Ψ t ) ◦ ( ε Ψ ) − ( ε Ψ ) ◦ ( f Ψ t ))Φ t . Consider a Z -grading on the algebra R such that the idempotents and arrows, except γ ,n − and γ ,n , are of degree 0 and the arrows γ ,n − and γ ,n are of degree 1. This gradinginduces a grading on Λ. We can define a grading on the direct summands of Q t ( t > Q ∗ is a graded resolution of the module R . Let t = t ′ + l (2 n − t ′ n − p ∈ X t and P [ x p ][ y p ] ≃ σ l ( P ) , where the module P appears in the formulafor Q t ′ as the module P [ i ,j ][ i ,j ] . We define the degree of the element e x p ⊗ e y p ∈ Q t in thefollowing way:1) if j , j
6∈ { n − , n } , thendeg( e x p ⊗ e y p ) = (cid:24) i + l ( n − − r (cid:25) − (cid:24) i − r (cid:25) ;2) if j
6∈ { n − , n } , j ∈ { n − , n } , thendeg( e x p ⊗ e y p ) = (cid:24) i + l ( n − r (cid:25) − (cid:24) i − r (cid:25) ;13) if j ∈ { n − , n } , j n − , n } , thendeg( e x p ⊗ e y p ) = (cid:24) i + l ( n − − r (cid:25) − (cid:24) i r (cid:25) ;4) if j , j ∈ { n − , n } , thendeg( e x p ⊗ e y p ) = (cid:24) i + l ( n − r (cid:25) − (cid:24) i r (cid:25) . Here we denote by ⌈ a ⌉ the smallest integer which is greater of equal to a . It is easy toshow that the differentials d Qt are actually of degree 0 for the grading introduced in 1)–4).Moreover, it is easy to check that, if we introduce the grading on the modules Bar t ( R ) in sucha way that deg( a ⊗· · ·⊗ a t +1 ) = t +1 P i =0 deg( a i ) for homogeneous elements a i ∈ R (0 i t +1),then Bar ∗ ( R ) becomes a graded resolution of R . In addition Φ t ( t >
0) is a homomorphismof graded modules. It is easy to check that ε Ψ ( b ) = deg( b ) b for b ∈ B R .It is clear that for elements f ∈ C ( R ) = Hom k ( R, R ) and g ∈ C t ( R ) = Hom k ( R ⊗ t , R ) thecomposition product f ◦ g is just a composition of f and g . Then (cid:0) ( ε Ψ ) ◦ ( f Ψ t ) (cid:1) Φ t = ε Ψ f for f ∈ Hom Λ ( Q t , R ), f d Qt = 0. Then it is easy to show that (cid:0) ( ε Ψ ) ◦ ( f Ψ t ) (cid:1) Φ t = , if f ∈ { f s , h s , p s } , n − P j = n − − m t ,m,j + u ,m,n − , if f = g s . (5.5)It follows from the formula (5.4) and the formula ( ε Ψ )( b ) = deg( b ) b ( b ∈ B R ) that (cid:0) ( f ◦ Ψ t ) ◦ ( ε Ψ ) (cid:1) Φ t ( e x p ⊗ e y p ) = deg( e x p ⊗ e y p ) f ( e x p ⊗ e y p )for p ∈ X t , f ∈ Hom Λ ( Q t , R ), f d Qt = 0. The assertion of proposition follows from thisformula and (5.5).Now we prove a theorem which combined with Proposition 1 and the results of [5] givesa full description of the algebra HH ∗ ( R ) as a Gerstenhaber algebra in all cases and as aBV-algebra in the case char k r . Theorem 3. If char k r , then HH ∗ ( R ) is a BV-algebra. In this case the BV-differential ∆ is defined by the following equalities: ∆( ε ) = 1 r , ∆( f s ) = ∆( g s ) = ∆( h s ) = ∆( p s ) = 0 , ∆( ε f s ) = f s r + [ f s , ε ] , ∆( ε g s ) = g s r + [ g s , ε ] , ∆( χ s ) = lr (cid:16) n f s − − p s − (cid:17) (2 | n, s = l (2 n − , ∆( χ s ) = 0 (2 n ) , ∆( ξ s ) = 2 lr h s − ( s = l (2 n − , ∆( ε ( q )0 ) = 0 . Suppose that char k | r . Then χ s and ξ s can be excluded from the set of generatorsand the Lie bracket is defined on the generators of HH ∗ ( R ) by the equalities (5.1) and theequalities [ f s , f s ] =[ f s , g s ] = [ f s , h s ] = [ f s , p s ] = [ g s , g s ] = [ g s , h s ]=[ g s , p s ] = [ h s , h s ] = [ p s , p s ] = 0 . Proof.
1) In this case HH ∗ ( R ) is a BV-algebra. Let us consider ˜ ε ∈ Hom Λ ( Q , R ) definedby the equality ˜ ε = r P i =1 ( u i, ,n − + u i, ,n ) r = r P i =1 ν − i ε θ i r . Then ˜ ε = ε in HH ∗ ( R ) and it is easy to show that ∆(˜ ε Ψ ) = r .Let us consider a Z -grading on the algebra R , which is induced by length. This gradinginduces a grading on Λ. We can define a grading on the direct summands of Q t ( t > Q ∗ is a graded resolution of the module R . Let t = t ′ + l (2 n − t ′ n − p ∈ X t and P [ x p ][ y p ] ≃ σ l ( P ) , where the module P appears in the formulafor Q t ′ as the module P [ i ,j ][ i ,j ] . Then we define the degree of the element e x p ⊗ e y p ∈ Q t by the formuladeg( e x p ⊗ e y p ) = l ( n − + a ( n −
1) + min( j , n − − min( j , n − . It is easy to show that the differentials d Qt are actually of degree 0 for this grading. Moreover,it is easy to check that, if we introduce a grading on the modules Bar t ( R ) in such a way thatdeg( a ⊗ · · · ⊗ a t +1 ) = t +1 P i =0 deg( a i ) for homogeneous elements a i ∈ R (0 i t + 1), thenBar ∗ ( R ) becomes a graded resolution of R . In addition Φ t and Ψ t ( t >
0) are homomorphismsof graded modules.Let M , N be Z -graded spaces. We say that a linear map ϕ : M → N is of degree q andwrite deg ϕ = q if deg ϕ ( m ) = deg m − q for any homogeneous element m ∈ M . Thus itis easy to show that a grading on R induces a grading on HH ∗ ( R ). Then direct inspectionshows that deg x = ( n − F ( x ) for x ∈ { ε , f s , g s , h s , p s , ξ s , χ s } . Note that any b ∈ B R ishomogeneous and satisfies the equalities deg ¯ b = n − − deg b and deg νb = deg b . In additionwe have h a, i = 0 for homogeneous a ∈ R such that deg a = n −
1. Suppose that x ∈ Ker δ νs and deg x = q . Let a i ∈ B R (1 i n − A = a ⊗ · · · ⊗ a n − . Then∆ i x ( A ) = X a ∈ B R h f ( a i ⊗ · · · ⊗ a n − ⊗ ¯ a ⊗ νa ⊗ · · · ⊗ νa i − ) , i a. Since deg f ( a i ⊗ · · · ⊗ a n − ⊗ ¯ a ⊗ νa ⊗ · · · ⊗ νa i − ) = deg A + deg ¯ a − q, nonzero coefficients can appear only for a ∈ B R such that deg a = deg A − q , i.e. deg(∆ i x ) = q . So deg(∆ x ) = deg x .If x ∈ { f s , g s , h s , p s } , then deg x = ( n − F ( x ) and there are no nonzero elements of suchdegree in HH t ( R ) for t < s . So ∆( f s ) = ∆( g s ) = ∆( h s ) = ∆( p s ) = 0. Then the equalitiesfor ∆( ε f s ) and ∆( ε g s ) follow from (2.1). We can calculate ∆( ε p s ) and ∆( ε h s ) using thesame formula. The formula for ∆( χ s ) in the case where 2 | n , char k n − ξ s ) in the case where 2 n , char k n − follow from [5, Lemma 1].15et now ˜ ξ s = r P i =1 ν − i ξ s θ i , ˜ χ s = r P i =1 ν − i χ s θ i . Then ξ s = ˜ ξ s r and χ s = ˜ χ s r in HH ∗ ( R ). Notethat the elements ˜ ξ s Ψ s and ˜ χ s Ψ s belong to C s ( R ) ν . In addition, if the elements ˜ ξ s Ψ s and˜ χ s Ψ s are defined for some field k , then they are defined for any field (for the same quiver).As it was said before the formulas for ∆( ξ s ) = ∆(˜ ξ s Ψ s )Φ s − r and ∆( χ s ) = ∆(˜ χ s Ψ s )Φ s − r are validfor a field k with zero characteristic.Let us introduce the notion of the standard basis for some modules. The standard basisfor Λ is the set B Λ = { a ⊗ b } a,b ∈ B R . If x, y ∈ Q , then the standard basis for P [ x ][ y ] is theset B Λ ∩ P [ x ][ y ] . And the standard basis for R ⊗ t is the set { a ⊗ · · · ⊗ a t } a ,...,a t ∈ B R . Thus wedefine the standard basis for Q t ( t >
0) and Bar t ( R ). Note that in all cases the definitionof the standard basis does not depend on the field. If the standard basis is defined for amodule M , we denote it by B M . We denote by Z B M the set of linear combinations withinteger coefficients of elements from B M . Note thata) if b ∈ Z B M and a ∈ B R , then ab, ba ∈ Z B M and the coefficients do not depend on thefield;b) if a ∈ Z B Q t , then d Qt − ( a ) ∈ Z B Q t − , D t ( a ) ∈ Z B Q t +1 and the coefficients do not dependon the field;c) if a ∈ B R , then ¯ a ∈ B R and νa ∈ B R and they do not depend on the field;d) if a, b ∈ Z B R , then h a, b i ∈ Z and it does not depend on the field.It follows from a) and b) that the elements of the matrices of Ψ t and Φ t in the standardbasis are integer and do not depend on the field. It follows from c) and d) that if the matrixof x : R ⊗ ( n +2) → R written in the standard basis consists of integer numbers, then the matrixof ∆( x ) : R ⊗ ( n +1) → R written in the standard basis consists of integer numbers which donot depend on the field. It follows from our arguments that the matrices of ∆( ˜ ξ s Ψ s )Φ s − : Q s − → R and ∆( ˜ χ s Ψ s )Φ s − : Q s − → R written in the standard bases consist of integernumbers which do not depend on the field. If x ∈ { ξ s , χ s } , then it follows from [5, Remark 5]that if the set of elements of degree ( n − F ( x ) is linear independent in Ker(Hom Λ ( d Qs − , R )),then it is linear independent in HH s − ( R ). The formulas for ∆( ˜ ξ s Ψ s )Φ s − and ∆( ˜ χ s Ψ s )Φ s − are written in terms of elements of Ker(Hom Λ ( d Qs − , R )) whose definitions do not depend onthe field. So these formulas remain true for any field.2) Since the elements χ s and ξ s appear only in the case CMD( n − , r ) = 1, they canbe excluded from the set of generators in this case. Let f, g ∈ { f s , g s , h s , p s } . Then thereare such x, y ∈ HH ∗ ( R ) ν ↑ that Θ νR ( x ) = f and Θ νR ( y ) = g . Since Θ νR is a homomorphismof Gerstenhaber algebras we have [ f, g ] = Θ νR ([ x, y ]). Since char k | r we have char k = 2 or2 | r . It is easy to check that in both cases elements f and g have even degree. In addition itfollows from the proof of [12, Lemma 3] and the formula (4.2) that if char k = 2 or 2 | r , thenfor any a ∈ Ker(Hom Λ ( d Q s +1 , R )) such that a = ν − aθ s +1 there is ¯ a ∈ Ker(Hom Λ ( d Q s +1 , R ))such that a = r − P i =0 ν − i ¯ aθ i s +1 . Since ¯ a ν = ¯ a in HH ∗ ( R ) by Corollary 2 of Lemma 1 we have a = r − X i =0 ν − i ¯ aθ i s +1 = r − X i =0 ¯ a ν i = r ¯ a = 0in HH s +1 ( R ). Consequently, Θ νR (HH s ( R ) ν ↑ ) = 0 for odd s . The element [ x, y ] has odddegree because elements x and y have even degree. Then [ f, g ] = Θ νR ([ x, y ]) = 0 and 2) isproved. 16 emark . It is easy to show that we can introduce the BV-structure on HH ∗ ( R ) in thecase where char k | r . For example, we can set ∆ equal to 0 on all generators of HH ∗ ( R ). References [1] M. Gerstenhaber,
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