Researches On The Hochschild Cohomology Groups Of Truncated Quiver Algebras

In this article,we mainly investigate the Lie module structure given on the Hochschild cohomology groups of truncated quiver algebras. We give the decomposition of odd or-der Hochschild cohomology groups related to the Lie module structure when the quiver is given by two loops with a vertex, also the semisimplicity of the first order Hochschild cohomology is described.Firstly, we will determine the graded Lie algebra structure of C*(Λ) using the mini-mal complex which plays a pivotal role in describing the semisimplicity of H H1(Λ) and the Lie module structure of H H2i+1(Λ). Thus we give an equivalent description on the semisimplicity of H H1(Λ) when quiver is arbitrary.Secondly, we firstly describe the Lie module structure of H H2i+1(A) over K(Q1||Q1) in the case of n loops quiver with a vertex. Afterwards, we prove that sln(K) can be embeded into K(Q1||Q1).Finally, we characterize the representation of the odd order Hochschild cohomology groups of truncated quiver algebras on sl2(K), and the result can be regraded as the generalized situation of the representation of the Hochschild cohomology groups of monomial algebras with radical square zero on sl2(K).