本项目以 Hochschild 上同调为主要工具来研究表示论与李理论中出现的一些重要代数类的形变理论与表示理论。 我们将利用A 无穷余代数与twisting 上链的技术构造代数的极小投射双模分解，使用比较映射的技术来计算若干重要代数类的Hochschild上同调的李代数结构，并藉此研究代数的形变与表示；特别地，希望将此方法应用到扭群代数的形变-Drinfeld Hecke代数上。我们将考虑Hopf代数胚的上同调中的twisted Poincare对偶与Batalin-Vilkovisky 结构。本项目将推进代数形变理论与Drinfeld Hecke代数表示理论的发展，增进我们对于Batalin-Vilkovisky结构的认识。

The aim of this project is to study the deformation theory and the representation theory of many important classes of algebras appearing in representation theory and Lie theory, using Hochschild cohomology as a main tool. We shall construct minimal projective bomodule resolutions of algebras using A-infinity coalgebras and twisting cochains, compute the Lie algebra structure over the Hochschild cohomology groups by constructing comparison morphisms, and study deformations and representation theory of algebras. In particular, we would like to apply this method to Drinfeld Hecke algebras which are deformations of skew group algebras. We shall consider twisted Poincare duality and Batalin-Vilkovisky structure in the cohomology theory of Hopf algebroids. This project will contribute to development of algebraic deformation theory and representation theory of Drinfeld Hecke algebras, and it will improve our understanding of Batalin-Vilkovisky structures.