On the first relative Hochschild cohomology

Abstract

In this paper we investigate the Lie algebra structure of the first relative Hochschild cohomology. Let $A,B$ be finite-dimensional basic $k$-algebras over an algebraically closed field of characteristic zero, such that $Q_B$ is a subquiver of $Q_A$. We show that if the complement of $Q_A$ by the arrows of $Q_B$ is a simple directed graph, then the first relative Hochschild cohomology ${\mathrm{HH}}^1(A|B)$ is a solvable Lie algebra. We also compute the Lie algebra structure of the first relative Hochschild cohomology for radical square zero algebras and for dual extension algebras of directed monomial algebras.

We have removed the section on the contracted fundamental group, which will instead appear in a forthcoming paper. We have also added two references. 19 pages