Let R=k(Q,I) be a finite-dimensional algebra over a field k determined by a bound quiver (Q,I). We show that if R is a simply connected right multipeak algebra which is chord-free and $\tilde{}$-free in the sense defined below then R has the separation property and there exists a preprojective component of the Auslander-Reiten quiver of the category prin(R) of prinjective R-modules. As a consequence we get in 4.6 a criterion for finite representation type of prin(R) in terms of the prinjective Tits quadratic...

We find some relations between module biprojectivity and module biflatness of Banach algebras $𝒜$ and $ℬ$ and their projective tensor product $𝒜\widehat{\otimes }ℬ$. For some semigroups $S$, we study module biprojectivity and module biflatness of semigroup algebras $l^1\left(S\right)$.

Let A be a special biserial algebra over an algebraically closed field. We show that the first Hohchshild cohomology group of A with coefficients in the bimodule A vanishes if and only if A is representation-finite and simply connected (in the sense of Bongartz and Gabriel), if and only if the Euler characteristic of Q equals the number of indecomposable non-uniserial projective-injective A-modules (up to isomorphism). Moreover, if this is the case, then all the higher Hochschild cohomology groups...

By an extension algebra of a finite-dimensional K-algebra A we mean a Hochschild extension algebra of A by the dual A-bimodule $Ho{m}_K(A,K)$. We study the problem of when extension algebras of a K-algebra A are symmetric. (1) For an algebra A= KQ/I with an arbitrary finite quiver Q, we show a sufficient condition in terms of a 2-cocycle for an extension algebra to be symmetric. (2) Let L be a finite extension field of K. By using a given 2-cocycle of the K-algebra L, we construct a 2-cocycle of the K-algebra...

We define a BV-structure on the Hochschild cohomology of a unital, associative algebra $A$ with a symmetric, invariant and non-degenerate inner product. The induced Gerstenhaber algebra is the one described in Gerstenhaber’s original paper on Hochschild-cohomology. We also prove the corresponding theorem in the homotopy case, namely we define the BV-structure on the Hochschild-cohomology of a unital $A_{\infty }$-algebra with a symmetric and non-degenerate $\infty$-inner product.

Let M be a closed orientable manifold of dimension dand ${𝒞}^{*}\left(M\right)$ be the usual cochain algebra on M with coefficients in a fieldk. The Hochschild cohomology of M, $H{H}^{*}({𝒞}^{*}\left(M\right);{𝒞}^{*}\left(M\right))$ is a graded commutative and associative algebra. The augmentation map $\epsilon :{𝒞}^{*}\left(M\right)\to 𝑘$ induces a morphism of algebras $I:H{H}^{*}({𝒞}^{*}\left(M\right);{𝒞}^{*}\left(M\right))\to H{H}^{*}({𝒞}^{*}\left(M\right);𝑘)$. In this paper we produce a chain model for the morphism I. We show that the kernel of I is a nilpotent ideal and that the image of I is contained in the center of $H{H}^{*}({𝒞}^{*}\left(M\right);𝑘)$, which is in general quite small. The algebra $H{H}^{*}({𝒞}^{*}\left(M\right);{𝒞}^{*}\left(M\right))$ is expected to be isomorphic...

This paper studies the Hochschild cohomology of finite-dimensional monomial algebras. If Λ = K/I with I an admissible monomial ideal, then we give sufficient conditions for the existence of an embedding of $K[x₁,...,x_r]/⟨x_a{x}_{b}fora\ne b⟩$ into the Hochschild cohomology ring HH*(Λ). We also introduce stacked algebras, a new class of monomial algebras which includes Koszul and D-Koszul monomial algebras. If Λ is a stacked algebra, we prove that $HH*\left(\Lambda \right)/\cong K[x₁,...,x_r]/⟨x_a{x}_{b}fora\ne b⟩$, where is the ideal in HH*(Λ) generated by the homogeneous nilpotent elements. In...

We determine the algebra structure of the Hochschild cohomology of the singular cochain algebra with coefficients in a field on a space whose cohomology is a polynomial algebra. A spectral sequence calculation of the Hochschild cohomology is also described. In particular, when the underlying field is of characteristic two, we determine the associated bigraded Batalin-Vilkovisky algebra structure on the Hochschild cohomology of the singular cochain on a space whose cohomology is an exterior algebra....