Bimodule Resolution Research Articles

Homotopy liftings and Hochschild cohomology of some twisted tensor products

The Hochschild cohomology of a tensor product of algebras is isomorphic to a graded tensor product of Hochschild cohomology algebras, as a Gerstenhaber algebra. A similar result holds when the tensor product is twisted by a bicharacter. We present new proofs of these isomorphisms, using Volkov’s homotopy liftings that were introduced for handling Gerstenhaber brackets expressed on arbitrary bimodule resolutions. Our results illustrate the utility of homotopy liftings for theoretical purposes.

Higher preprojective algebras, Koszul algebras, and superpotentials

In this article we study higher preprojective algebras, showing that various known results for ordinary preprojective algebras generalize to the higher setting. We first show that the quiver of the higher preprojective algebra is obtained by adding arrows to the quiver of the original algebra, and these arrows can be read off from the last term of the bimodule resolution of the original algebra. In the Koszul case, we are able to obtain the new relations of the higher preprojective algebra by differentiating a superpotential and we show that when our original algebra is$d$-hereditary, all the relations come from the superpotential. We then construct projective resolutions of all simple modules for the higher preprojective algebra of a$d$-hereditary algebra. This allows us to recover various known homological properties of the higher preprojective algebras and to obtain a large class of almost Koszul dual pairs of algebras. We also show that when our original algebra is Koszul there is a natural map from the quadratic dual of the higher preprojective algebra to a graded trivial extension algebra.

Gerstenhaber brackets on Hochschild cohomology of general twisted tensor products

We present techniques for computing Gerstenhaber brackets on Hochschild cohomology of general twisted tensor product algebras . These techniques involve twisted tensor product resolutions and are based on recent results on Gerstenhaber brackets expressed on arbitrary bimodule resolutions.

$A_{\infty}$-coderivations and the Gerstenhaber bracket on Hochschild cohomology

We show that Hochschild cohomology of an algebra over a field is a space of infinity coderivations on an arbitrary projective bimodule resolution of the algebra. The Gerstenhaber bracket is the graded commutator of infinity coderivations. We thus generalize, to an arbitrary resolution, Stasheff’s realization of the Gerstenhaber bracket on Hochschild cohomology as the graded commutator of coderivations on the tensor coalgebra of the algebra.

Group-Twisted Alexander-Whitney and Eilenberg-Zilber Maps

We define group-twisted Alexander-Whitney and Eilenberg-Zilber maps for converting between bimodule resolutions of skew group algebras. These algebras are the natural semidirect products recording actions of finite groups by automorphisms. The group-twisted chain maps allow us to transfer information between resolutions for use in homology theories, for example, in those governing deformation theory. We show how to translate in particular from the default (but often cumbersome) reduced bar resolution to a more convenient twisted product resolution. This provides a more universal approach to some known results classifying PBW deformations.

Hochschild Cohomology for Algebras of Dihedral Type. VII. The Family D(3R)

The Hochschild cohomology groups for algebras of dihedral type which are contained in the family D(3R) (from the famous K.Erdmann’s classification) are calculated. In the calculation, a construction of the minimal bimodule resolution for algebras from the family under discussion, that is defined in the present paper, is used.

Hochschild Cohomology for Algebras of Semidihedral Type. VIII. The Family SD(2B)1

The Hochschild cohomology groups for algebras of semidihedral type, that are contained in the family SD(2B)1 (from the famous K. Erdmann’s classification), are computed. In the calculation, a construction of the minimal bimodule resolution for algebras from the family under discussion, that is defined in the present paper, is used.

Gerstenhaber Bracket on the Hochschild Cohomology via An Arbitrary Resolution

AbstractWe prove formulas of different types that allow us to calculate the Gerstenhaber bracket on the Hochschild cohomology of an algebra using some arbitrary projective bimodule resolution for it. Using one of these formulas, we give a new short proof of the derived invariance of the Gerstenhaber algebra structure on Hochschild cohomology. We also give some new formulas for the Connes differential on the Hochschild homology that lead to formulas for the Batalin–Vilkovisky (BV) differential on the Hochschild cohomology in the case of symmetric algebras. Finally, we use one of the obtained formulas to provide a full description of the BV structure and, correspondingly, the Gerstenhaber algebra structure on the Hochschild cohomology of a class of symmetric algebras.

On Derived Equivalence of Algebras of Semidihedral Type with Two Simple Modules

The Hochschild cohomology groups of degrees not exceeding 3 are computed for algebras of semidihedral type that form the family SD(2B)1 (from the famous K. Erdmann’s classification). In the calculation, the beforehand construction of the initial part of the minimal projective bimodule resolution is used for algebras from the family under discussion. The obtained results imply that the algebras from the families SD(2B)1 and SD(2B)2 with the same parameters in defining relations are not derived equivalent.

Hochschild homology and cohomology of down–up algebras

We present a detailed computation of the cyclic and the Hochschild homology and cohomology of generic and 3-Calabi–Yau homogeneous down–up algebras. This family was defined by Benkart and Roby in [3] in their study of differential posets. Our calculations are completely explicit, by making use of the Koszul bimodule resolution and some arguments similar to those used in [13] to compute the Hochschild cohomology of Yang–Mills 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.

Internally Calabi\u2013Yau algebras and cluster-tilting objects

We describe what it means for an algebra to be internally d-Calabi–Yau with respect to an idempotent. This definition abstracts properties of endomorphism algebras of (d-1)-cluster-tilting objects in certain stably (d-1)-Calabi–Yau Frobenius categories, as observed by Keller–Reiten. We show that an internally d-Calabi–Yau algebra satisfying mild additional assumptions can be realised as the endomorphism algebra of a (d-1)-cluster-tilting object in a Frobenius category. Moreover, if the algebra satisfies a stronger ‘bimodule’ internally d-Calabi–Yau condition, this Frobenius category is stably (d-1)-Calabi–Yau. We pay special attention to frozen Jacobian algebras; in particular, we define a candidate bimodule resolution for such an algebra, and show that if this complex is indeed a resolution, then the frozen Jacobian algebra is bimodule internally 3-Calabi–Yau with respect to its frozen idempotent. These results suggest a new method for constructing Frobenius categories modelling cluster algebras with frozen variables, by first constructing a suitable candidate for the endomorphism algebra of a cluster-tilting object in such a category, analogous to Amiot’s construction in the coefficient-free case.

On the Hochschild cohomology ring of integral cyclic algebras

We determine the ring structure of the Hochschild cohomology HH*(Γ) of an integral cyclic algebra Γ by giving a projective bimodule resolution of Γ and calculating cup product by means of a diagonal approximation map.

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.

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.

Periodicity of self-injective algebras of polynomial growth

Let A be an indecomposable representation-infinite tame finite-dimensional algebra of polynomial growth over an algebraically closed field. We prove that A is a periodic algebra with respect to action of the bimodule syzygy operator if and only if A is Morita equivalent to a socle deformation of an orbit algebra Bˆ/G where Bˆ is the repetitive category of a tubular algebra B and G is an admissible infinite cyclic automorphism group of Bˆ. The main contribution in the paper is to prove the sufficiency part of this equivalence. It is known that every orbit algebra Bˆ/G of a tubular algebra B admits a presentation as an orbit algebra T(B)(r)/H of an r-fold trivial extension algebra T(B)(r) of B with respect to free action of a finite cyclic automorphism group H of T(B)(r). A significant part of the paper is devoted to explicit descriptions of the minimal projective bimodule resolutions of properly chosen ten exceptional self-injective algebras of polynomial growth and showing that all of them are periodic algebras. Then we deduce the periodicity of all algebras socle equivalent to the orbit algebras Bˆ/G=T(B)(r)/H of tubular algebras B from the periodicity of these ten exceptional algebras, using invariance of periodicity for finite Galois coverings and derived equivalences of algebras.

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.

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.

Projective resolutions of associative algebras and ambiguities

The aim of this article is to give a method to construct bimodule resolutions of associative algebras, generalizing Bardzell's well-known resolution of monomial algebras. We stress that this method leads to concrete computations, providing thus a useful tool for computing invariants associated to the considered algebras. We illustrate how to use it by giving several examples in the last section of the article. In particular we give necessary and sufficient conditions for noetherian down–up algebras to be 3-Calabi–Yau.

Hochschild Cohomology for Algebras of Semidihedral Type. IV. The Cohomology Algebra for the Family SD(2ℬ)2(k, t, c) in the Case c = 0

The paper continues the previous author’s paper in which the bimodule resolution was constructed for algebras of semidihedral type from the family SD(2ℬ)2. In the present paper, using this resolution the multiplicative structure of the Hochschild cohomology algebra for algebras in the mentioned family is described over a base field of characteristic 2 under an additional assumption that the parameter c contained in the defining relations of the algebras is equal to zero.