The Batalin-Vilkovisky Algebraic Structure On The Hochschild Cohomology Of Two Kinds Of Koszul Algebras

The theory of Hochschild homology and cohomology of associative algebra play an important role in the representation theory of algebra and non-commutative geometry.This paper mainly study the Gerstenhaber algebraic structure and Batalin-Vilkovisky algebraic structure on Hochschild cohomology of two kinds of important finite-dimensional algebras.The first kind of algebra considered in this paper is the quantum exterior algebra with two variables,which is used by Buchweitz et al to give a counterexample for Happel's problem.This is a class of Frobenius Koszul algebras with semisimple Nakayama automorphism.The second kind of algebra is the tame Hecke algebra of type A.This is a class of special biserial symmetric Koszul algebras.For the two kinds of algebras,we firstly construct two chain maps between the minimal projective bimodule resolution and the reduced Bar resolution by the weak self-homotopy.Secondly,by using the chain maps and the non-degenerate bilinear form constructed by Tradler and Volkov,we get the Batalin-Vilkovisky operator.Therefore,we give the Gerstenhaber algebraic structure and the Batalin-Vilkovisky algebraic structure on Hochschild cohomology of the two kinds of algebras.