On the first Hochschild cohomology of finite dimensional quiver algebras under gluing idempotents

Abstract

We compare the Lie algebra structures of the first Hochschild cohomology groups of a quiver algebra $A$ and a radical embedding $B$ obtained by gluing two idempotents of $A$. Under a mild assumption, we show that the first Hochschild cohomology groups of $A$ and $B$ are either isomorphic as Lie algebras or they differ by a one-dimensional Lie ideal. In particular, in the case of stable equivalences obtained by gluing a source and a sink vertex, we prove that either the first Hochschild cohomology groups of $A$ and $B$ are isomorphic or $H{H}^1(B)$ is a central extension of $H{H}^1(A)$ by a one-dimensional ideal. As a consequence, we obtain a new invariant under stable equivalences induced by gluing a source and a sink. We also compare the dimensions of $H{H}^1(A)$ and $H{H}^1(B)$, as well as the centers of $A$ and $B$, when gluing two arbitrary idempotents.

arXiv commentThe paper has been rewritten to generalize the main results from monomial algebras to quiver algebras. The sections on fundamental groups and Hochschild cohomology groups in higher degrees have been omitted. Consequently the title has been changed