The Hochschild Cohomology Groups Of Two Special Algebras

The Hochschild cohomology theory was introduced by Hochschild in 1945, and developed by Carten and Eilenberg. It plays an important role in many mathematical branches, such as the representation thoery of algebras, Lie algebras, algebraic topology, algebraic geometry, and so on. Generally, the Hochschild cohomology groups are closely interconnected with the structure of an associative algebra. Specifically, the 0-th Hochschild cohomology group is equal to the center of algebra; the 1-th Hochschild cohomology group is the outer derivations of the associative algebra; the 2-th and 3-th Hochschild cohomology groups have some important applications in the formal deforma-tion theory of algebras. Therefore, it is of interest to compute the Hochschild cohomology groups for some specail algebras, such as the finite dimensional hereditary algebras, the incidence algebras, the algebras with narrow quivers, the truncated algebras, the algebras with some zero relarion, and so on. From these well-known results, we find that it is difficult to compute the Hochschild cohomology groups if the Gabriel quiver of an algebra has parallel paths or directed cycles. In this thesis, we proceed with the study of Hochschild co-homology groups for two sepecial quiver algebras. This thesis is divided into three parts:In the first chapter, we introduce some background and development of Hochschild cohomology theory, and some necessary notions and our main work.In chapter 2, we compute the Hochschild cohomology group of of factor algebras of parallel path algebras by constructing the minimal projective res-olution of algebras, where the parallel path algebra means that its Gabriel quiver consists of two parallel paths. In this chapter, we compute completely the Hochschild cohomology groups of the factor algebras of paralell path of length 2 up to isomorphism.In the third chapter, we investigate the Hochschild cohomology groups for one-point extensions of basic cycles by some known results about truncated basic cycle algebras and one-point extension of algebras. Finally, we discuss the Hochschild cohomology groups for a one-point extensions of 2-truncated basic cycles by a simple module.