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 $HH^1(B)$ is a central extension of $HH^1(A)$ 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 $HH^1(A)$ and $HH^1(B)$, as well as the centers of $A$ and $B$, when gluing two arbitrary idempotents.