Hochschild Cohomology Groups Research Articles

Gerstenhaber Algebra Structure on the Hochschild Cohomology of Quadratic String Algebras

We describe the Gerstenhaber algebra structure on the Hochschild cohomology HH∗(A) when A is a quadratic string algebra. First we compute the Hochschild cohomology groups using Barzdell’s resolution and we describe generators of these groups. Then we construct comparison morphisms between the bar resolution and Bardzell’s resolution in order to get formulae for the cup product and the Lie bracket. We find conditions on the bound quiver associated to string algebras in order to get non-trivial structures.

Cyclic cohomology and Chern–Connes pairing of some crossed product algebras

We compute the cyclic and Hochschild cohomology groups for the algebras Aθalg⋊Z3,Aθalg⋊Z4 and Aθalg⋊Z6. We also compute the partial Chern–Connes index table for each of these algebras.

Hochschild Cohomology for Algebras of Semidihedral Type. VI. The Family S D 2 ℬ 2 $$ SD{\left(2\mathrm{\mathcal{B}}\right)}_2 $$ in Characteristic Different from 2

The Hochschild cohomology groups for algebras of semidihedral type that lie in the family $$ SD{\left(2\mathrm{\mathcal{B}}\right)}_2 $$ (in the famous K.Erdmann’s classification) over an algebraically closed field with characteristic different from two are computed. The calculation, relies upon the minimal projective bimodule resolution for algebras from the above family that was constructed in the previous author’s paper.

Derivations and automorphisms of the positive part of the two-parameter quantum group U r,s (B 3)

We compute the derivations of the positive part of the two-parameter quantum group U r,s (B 3) and show that the Hochschild cohomology group of degree 1 of this algebra is a three-dimensional vector space over the base field C. We also compute the groups of (Hopf) algebra automorphisms of the augmented two-parameter quantized enveloping algebra Ǔ r,s ≥0(B 3).

Hochschild Cohomology of Algebras of Semidihedral Type. V. The Family S D 3 K $$ SD\left(3\mathcal{K}\right) $$

The Hochschild cohomology groups are calculated for the algebras of semidihedral type that form the family $$ SD\left(3\mathcal{K}\right) $$ (from the famous K. Erdmann’s classification). In the calculation, the bimodule resolution previously constructed for the algebras belonging to the family under discussion is used.

Exact sequences, Hochschild cohomology, and the Lie module structure over the M-relative center

In this article, we present actions by central elements on Hochschild cohomology groups with arbitrary bimodule coefficients, as well as an interpretation of these actions in terms of exact sequences. Since our construction utilizes the monoidal structure that the category of bimodules possesses, we will further recognize that these actions are compatible with monoidal functors and thus, as a consequence, are invariant under Morita equivalences. By specializing the bimodule coefficients to the underlying algebra itself, our efforts in particular yield a description of the degree-(n,0)-part of the Lie bracket in Hochschild cohomology, and thereby close a gap in earlier work by S. Schwede.

Hochschild (co)homology of Yoneda algebras of reconstruction algebras of type A 1

The reconstruction algebra is a generalization of the preprojective algebra, and plays important roles in algebraic geometry and commutative algebra. We consider the homological property of this class of algebras by calculating the Hochschild homology and Hochschild cohomology. Let Λt be the Yoneda algebra of a reconstruction algebra of type A1 over a field k. In this paper, a minimal projective bimodule resolution of Λt is constructed, and the k-dimensions of all Hochschild homology and cohomology groups of Λt are calculated explicitly.

On Hochschild Cohomology of a Self-injective Special Biserial Algebra Obtained by a Circular Quiver with Double Arrows

We calculate the dimensions of the Hochschild cohomology groups of a self-injective special biserial algebra Λ s obtained by a circular quiver with double arrows. Moreover, we give a presentation of the Hochschild cohomology ring modulo nilpotence of Λ s by generators and relations. This result shows that the Hochschild cohomology ring modulo nilpotence of Λ s is finitely generated as an algebra.

Module and Hochschild cohomology of certain semigroup algebras

We study the relation between the module and Hochschild cohomology groups of Banach algebras.We show that, for every commutative Banach A-A-bimodule X and every k ∈ N, the seminormed spaces HAk (A,X*) and Hk(A /J,X*) are isomorphic, where J is a specific closed ideal of A. As an example, we show that, for an inverse semigroup S with the set of idempotents E, where l1(E) acts on l1(S) by multiplication on the right and trivially on the left, the first module cohomology \(H_{{\ell ^1}\left( E \right)}^1\) (l1(S), l1(GS)(2n+1)) is trivial for each n ∈ N, where GS is the maximal group homomorphic image of S. Also, the second module cohomology \(H_{{\ell ^1}\left( E \right)}^2\) (l1(S), l1(GS)(2n+1)) is a Banach space.

On Simple Connectedness of Minimal Representation-Infinite Algebras

Let A be a connected minimal representation-infinite algebra over an algebraically closed field k. In this paper, we investigate the simple connectedness and strong simple connectedness of A. We prove that A is simply connected if and only if its first Hochschild cohomology group H1(A) is trivial. We also give some equivalent conditions of strong simple connectedness of an algebra A.

Hochschild Cohomology for Algebras of Dihedral Type. IV. The family D(2B)(k, s, 0)

The Hochschild cohomology groups for algebras of dihedral type that lie in the family D(2B)(k, s, c) (in the famous K. Erdmann’s classification) are computed in the case where the parameter c included in the defining relations of algebras from this family is equal to zero. The calculation relies upon the construction of a bimodule resolution for algebras from the above family. Bibliography: 26 titles.

On the First Hochschild Cohomology Group of a Cluster-Tilted Algebra

Given a cluster-tilted algebra B, we study its first Hochschild cohomology group HH1(B) with coefficients in the B-B-bimodule B. If C is a tilted algebra such that B is the relation-extension of C, then we show that if B is tame, then HH1(B) is isomorphic, as a k-vector space, to the direct sum of \({\text {HH}}^{1}(C)\) with \(k^{n_{B,C}}\), where nB,C is an invariant linking the bound quivers of B and C. In the representation-finite case, HH1(B) can be read off simply by looking at the quiver of B.

Hochschild cohomology rings of Temperley-Lieb algebras

The authors first construct an explicit minimal projective bimodule resolution (ℙ, δ) of the Temperley-Lieb algebra A, and then apply it to calculate the Hochschild cohomology groups and the cup product of the Hochschild cohomology ring of A based on a comultiplicative map Δ: ℙ → ℙ ⊗ A ℙ. As a consequence, the authors determine the multiplicative structure of Hochschild cohomology rings of both Temperley-Lieb algebras and representation-finite q-Schur algebras under the cup product by giving an explicit presentation by generators and relations.

On the Decomposition of the Hochschild Cohomology Group of a Monomial Algebra Satisfying a Separability Condition

In this paper, we consider the finite connected quiver Q having two subquivers Q (1) and Q (2) with . Suppose that Q (i) is not a subquiver of Q (j) where {i, j} = {1, 2}. For a monomial algebra Λ =kQ/I obtained by the quiver Q, when the associated sequence of paths given by Minsharp <(I) satisfies a certain separability condition, we propose the method so that we easily construct a minimal projective resolution of Λ as a right Λ e -module and calculate the Hochschild cohomology group of Λ.

Spectral sequences for the cohomology rings of a smash product

Stefan and Guichardet have provided Lyndon–Hochschild–Serre type spectral sequences which converge to the Hochschild cohomology and Ext groups of a smash product. We show that these spectral sequences carry natural multiplicative structures, and that these multiplicative structures can be used to calculate the cup product on Hochschild cohomology and the Yoneda product on an Ext algebra.

Derivations of a parametric family of subalgebras of the Weyl algebra

An Ore extension over a polynomial algebra F[x] is either a quantum plane, a quantum Weyl algebra, or an infinite-dimensional unital associative algebra Ah generated by elements x,y, which satisfy yx−xy=h, where h∈F[x]. When h≠0, the algebra Ah is subalgebra of the Weyl algebra A1 and can be viewed as differential operators with polynomial coefficients. This paper determines the derivations of Ah and the Lie structure of the first Hochschild cohomology group HH1(Ah)=DerF(Ah)/InderF(Ah) of outer derivations over an arbitrary field. In characteristic 0, we show that HH1(Ah) has a unique maximal nilpotent ideal modulo which HH1(Ah) is 0 or a direct sum of simple Lie algebras that are field extensions of the one-variable Witt algebra. In positive characteristic, we obtain decomposition theorems for DerF(Ah) and HH1(Ah) and describe the structure of HH1(Ah) as a module over the center of Ah.

Algebra endomorphisms and derivations of some localized down-up algebras

We study algebra endomorphisms and derivations of some localized down-up algebras A𝕊(r + s, -rs). First, we determine all the algebra endomorphisms of A𝕊(r + s, -rs) under some conditions on r and s. We show that each algebra endomorphism of A𝕊(r + s, -rs) is an algebra automorphism if rmsn = 1 implies m = n = 0. When r = s-1 = q is not a root of unity, we give a criterion for an algebra endomorphism of A𝕊(r + s, -rs) to be an algebra automorphism. In either case, we are able to determine the algebra automorphism group for A𝕊(r + s, -rs). We also show that each surjective algebra endomorphism of the down-up algebra A(r + s, -rs) is an algebra automorphism in either case. Second, we determine all the derivations of A𝕊(r + s, -rs) and calculate its first degree Hochschild cohomology group.

Derivations on Toeplitz Algebras

AbstractLet H2(Ω) be the Hardy space on a strictly pseudoconvex domain Ω ⊂ ℂn, and let A ⊂ L∞(∂Ω) denote the subalgebra of all L∞-functions ƒ with compact Hankel operator Hƒ. Given any closed subalgebra B ⊂ A containing C(Ω), we describe the first Hochschild cohomology group of the corresponding Toeplitz algebra 𝒯(B) ⊂ B(H2(Ω). In particular, we show that every derivation on 𝒯(A) is inner. These results are new even for n = 1, where it follows that every derivation on T(H∞ +C) is inner, while there are non-inner derivations on T(H∞ + C(∂ℝn)) over the unit ball Bn in dimension n &gt; 1.

Hochschild cohomology for a class of some self-injective special biserial algebras of rank four

In this paper, we construct an explicit minimal projective bimodule resolution of a self-injective special biserial algebra AT (T≥0) whose Grothendieck group is of rank 4. As the main result, we determine the dimension of the Hochschild cohomology group HHi(AT) of AT for i≥0, completely. Moreover we give a presentation of the Hochschild cohomology ring modulo nilpotence HH⁎(AT)/NAT of AT by generators and relations in the case where T=0.

Structure of Hochschild cohomology of path algebras and differential formulation of Euler’s polyhedron formula

This article studies the Lie algebra Der(kΓ) of derivations on the path algebra kΓ of a quiver Γ and the Lie algebra on the first Hochschild cohomology group HH1(kΓ). We relate these Lie algebras to the algebraic and combinatorial properties of the path algebra. Characterizations of derivations on a path algebra are obtained, leading to a canonical basis of Der(kΓ) and its Lie algebra properties. Special derivations are associated to the vertices, arrows and faces of a quiver, and the concepts of a connection matrix and boundary matrix are introduced to study the relations among these derivations, concluding that the space of edge derivations is the direct sum of the spaces of the vertex derivations and the face derivations, while the dimensions of the latter spaces are the largest possible. By taking dimensions, this relation among spaces of derivations recovers Euler’s polyhedron formula. This relation also leads a combinatorial construction of a canonical basis of the Lie algebra HH1(kΓ), together with a new semidirect sum decomposition of HH1(kΓ).