# On the Lie algebra structure of $HH^1(A)$ of a finite-dimensional algebra $A$

## TL;DR

This paper investigates the Lie algebra structure of the first Hochschild cohomology of finite-dimensional algebras, revealing conditions under which it is solvable or isomorphic to sl_2(k), based on properties of the Ext-quiver.

## Contribution

It establishes new criteria linking the Ext-quiver properties of an algebra to the solvability and simplicity of its Hochschild cohomology Lie algebra.

## Key findings

- If the Ext-quiver is a simple directed graph, then HH^1(A) is solvable.
- If the Ext-quiver has no loops and at most two parallel arrows, and HH^1(A) is simple, then char(k) ≠ 2 and HH^1(A) ≅ sl_2(k).
- Investigates symmetric algebras with a quiver having a vertex with a single loop.

## Abstract

Let $A$ be a split finite-dimensional associative unital algebra over a field. The first main result of this note shows that if the Ext-quiver of $A$ is a simple directed graph, then $HH^1(A)$ is a solvable Lie algebra. The second main result shows that if the Ext-quiver of $A$ has no loops and at most two parallel arrows in any direction, and if $HH^1(A)$ is a simple Lie algebra, then char(k) is not equal to $2$ and $HH^1(A)\cong$ $sl_2(k)$. The third result investigates symmetric algebras with a quiver which has a vertex with a single loop.