The first Hochschild cohomology as a Lie algebra

TL;DR

This paper investigates conditions under which the first Hochschild cohomology of finite dimensional algebras forms a solvable Lie algebra, providing criteria based on the Ext-quiver structure.

Contribution

It establishes new sufficient conditions for the solvability of Hochschild cohomology Lie algebras using quiver properties, applicable to broad algebra classes.

Findings

Hochschild cohomology is solvable if the quiver has no loops or parallel arrows.

Provides verifiable criteria for solvability in quivers with loops.

Shows solvability for large classes of algebras, including tame blocks and quantum complete intersections.

Abstract

In this paper we study sufficient conditions for the solvability of the first Hochschild cohomology of a finite dimensional algebra as a Lie algebra in terms of its Ext-quiver in arbitrary characteristic. In particular, we show that if the quiver has no parallel arrows and no loops then the first Hochschild cohomology is solvable. For quivers containing loops, we determine easily verifiable sufficient conditions for the solvability of the first Hochschild cohomology. We apply these criteria to show the solvability of the first Hochschild cohomology space for large families of algebras, namely, several families of self-injective tame algebras including all tame blocks of finite groups and some wild algebras including most quantum complete intersections.