Batalin-Vilkovisky Structure on Hochschild Cohomology of Self-Injective Quadratic Monomial Algebras

We give a complete description of the Batalin-Vilkovisky algebra structure on Hochschild cohomology of the self-injective quadratic monomial algebras.

BV Structure on Hochschild Cohomology of Quantum Exterior Algebra with Two Variables

Let Ag=k(x,y)/(x^(2),xy+qyx,y^(2))over a field k.We give a clear character-ization of the Batalin-Vilkovisky algebraic structure on Hochschild cohomology of A_(q)for any q≠O,and the Gerstenhaber algebraic structure on Hochschild cohomology of A_(q)for q=0.

Hochschild cohomology of truncated quiver algebras

For a truncated quiver algebra over a field of an arbitrary characteristic, its Hochschild cohomology is calculated. Moreover, it is shown that its Hochschild cohomology algebra is finitedimensional if and only if its global dimension is finite if and only if its quiver has no oriented cycles.

Hochschild cohomology of special biserial algebras

Based on a four-term exact sequence,the formulae on the dimensions of the first and the second Hochschild cohomology groups of special biserial algebras with normed bases are obtained in terms of combinatorics.

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 → PAP. 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.

Graded Hochschild Cohomology of a Path Algebra with Oriented Cycles

The aim of this paper is to characterize the first graded Hochschild cohomology of a hereditary algebra whose Gabriel quiver is admitted to have oriented cycles. The interesting conclusion we have obtained shows that the standard basis of the first graded Hochschild cohomology depends on the genus of a quiver as a topological object. In this paper, we overcome the limitation of the classical Hochschild cohomology for hereditary algebra where the Gabriel quiver is assumed to be acyclic. As preparation, we first investigate the graded differential operators on a path algebra and the associated graded Lie algebra.

Hochschild cohomology of Beilinson algebra of exterior algebra

Let An be the Beilinson algebra of exterior algebra of an n-dimensional vector space, which is derived equivalent to the endomorphism algebra Endox （T） of a tilting complex T = II^ni=0^Ox （i） of coherent COx-modules over a projective scheme X = P^nk. In this paper we first construct a minimal projective bimodule resolution of An, and then apply it to calculate k-dimensions of the Hochsehild cohomology groups of An in terms of parallel paths. Finally, we give an explicit description of the cup product and obtain a Gabriel presentation of Hochschild cohomology ring of An. As a consequence, we provide a class of algebras of finite global dimension whose Hochschild cohomology rings have non-trivial multiplicative structures.

Batalin-Vilkovisky algebra structures on Hochschild cohomology of generalized Weyl algebras

We devote to the calculation of Batalin–Vilkovisky algebra structures on the Hochschild cohomology of skew Calabi–Yau generalized Weyl algebras.We first establish a Van den Bergh duality at the level of complex.Then based on the results of Solotar et al.,we apply Kowalzig and Krähmer's method to the Hochschild homology of generalized Weyl algebras,and translate the homological information into cohomological one by virtue of the Van den Bergh duality,obtaining the desired Batalin–Vilkovisky algebra structures.Finally,we apply our results to quantum weighted projective lines and Podleśquantum spheres,and the Batalin–Vilkovisky algebra structures for them are described completely.

The Hochschild Cohomology of the Chinese Monoid Algebra

In this work we find the groups of Hochschild cohomologies for the Chinese monoid algebra and derive its Hilbert and Poincaréseries.In order to obtain this result we construct the Anick resolution via the algebraic discrete Morse theory and Grobner-Shirshov basis for the Chinese monoid.

G-Algebra Structure on the Higher Order Hochschild Cohomology H*_(S)^(2)(A,A)

We present a deformation theory associated to the higher Hochschild coho-mology H*_(S)^(2)(A,A).We also study a G-algebra structure associated to this deformation theory.

The Hochschild Cohomology Ring of Temperley–Lieb Algebras Revisited

We determine the Gerstenhaber algebra structure on the Hochschild cohomology ring of Temperley–Lieb algebras in this paper.

Hochschild cohomology of Zn-Galois coverings of an algebra

We consider the Zn-Galois covering An of the algebra A introduced by F. Xu [Adv. Math., 2008, 219： 1872-1893]. We calculate the dimensions of all Hochschild cohomology groups of An and give the ring structure of the Hochschild cohomology ring modulo nilpotence. As a conclusion, we provide a class of counterexamples to Snashall-Solberg＇s conjecture.

Hochschild cohomology ring modulo nilpotence of a one point extension of a quiver algebra defined by two cycles and a quantum-like relation

We consider a one point extension algebra B of a quiver algebra Aq over a field k defined by two cycles and a quantum-like relation depending on a nonzero element q in k. We determine the Hochschild cohomology ring of B modulo nilpotence and show that if q is a root of unity, then B is a counterexample to Snashall-Solberg＇s conjecture.

On the Hochschild Cohomology and Homology of Endomorphism Algebras of Exceptional Sequences over Hereditary Algebras

In this paper, let A be a finite dimensional associative algebra over an algebraically closed field h, modA be the category of finite dimensional left A-module and X1,X2,... ,X2, in modA be a complete exceptional sequence, and let E be the endomorphism algebra of X1, X2,..., Xn. We study the global dimension of E, and calculate the Hochschild cohomology and homology groups of E.

一类Gerstenhaber代数的构造

For any K-algebra A,based on Hochschild complex and Hochschild coho-mology of A,we construct a new Gerstenhaber algebra,and give Gerstenhaber algebra epimorphism from the new Gerstenhaber algebra to the Gerstenhaber algebra of the Hochschild cohomology of A.

Hochschild (Co)homology of a Class of Nakayama Algebras

Let A = kQ/I be a finite-dimensional Nakayama algebra, where Q is an Euclidean diagram An for some n with cyclic orientation, and I is an admissible ideal generated by a single monomial relation. In this note we determine explicitly all the Hochschild homology and cohomology groups of A based on a detailed description of the Bardzell complex. Moreover, the cyclic homology of A can be calculated in the case that the underlying field is of characteristic zero.

A Note on "Modules, Comodules, and Cotensor Products over Frobenius Algebras"

This is a note on Abrams＇ paper ＂Modules, Comodules, and Cotensor Products over Frobenius Algebras, Journal of Algebras＂ （1999）. With the application of Frobenius coordinates developed recently by Kadison, one has a direct proof of Abrams＇ characterization for Frobenius algebras in terms of comultiplication （see L. Kadison （1999））. For any Frobenius algebra, by using the explicit comultiplication, the explicit correspondence between the category of modules and the category of comodules is obtained. Moreover, with this we give very simplified proofs and improve Abrams＇ results on the Hom functor description of cotensor functor.

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 Ur,s（B3） 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 Ur,s（B3）.

On Some Problems Studied by R. V. Kadison

Several problems studied by professor R. V. Kadison are shown to be closely related. The problems were originally formulated in the contexts of homomorphisms of C*-algebras, cohomology of von Neumann algebras and perturbations of C*-algebras. Recent research by G. Pisier has demonstrated that all of the problems considered are related to the question of whether all C*-algebras have finite length.

Derivations of the Positive Part of the Two-parameter Quantum Group of Type G_(2)

In this paper,we compute the derivations of the positive part of the two-parameter quantum group of type G_(2) by embedding it into a quantum torus.We also show that the first Hochschild cohomology group of this algebra is a two-dimensional vector space over the complex field.