# On solvability of the first Hochschild cohomology of a finite-dimensional algebra

## TL;DR

This paper develops a method to determine when the first Hochschild cohomology of a finite-dimensional algebra is solvable, providing explicit formulas and structural descriptions, especially for tame or finite representation type algebras.

## Contribution

It introduces a general approach to analyze the solvability of HH^1(A) and describes its structure for specific algebra classes, answering an open question.

## Key findings

- Criteria for solvability of HH^1(A) as a Lie algebra.
- Explicit decomposition of HH^1(A) for tame or finite type algebras.
- Formula for counting copies of sl_2 in the decomposition.

## Abstract

For an arbitrary finite-dimensional algebra $A$, we introduce a general approach to determining when its first Hochschild cohomology ${\rm HH}^1(A)$, considered as a Lie algebra, is solvable. If $A$ is moreover of tame or finite representation type, we are able to describe ${\rm HH}^1(A)$ as the direct sum of a solvable Lie algebra and a sum of copies of $\mathfrak{sl}_2$. We proceed to determine the exact number of such copies, and give an explicit formula for this number in terms of certain chains of Kronecker subquivers of the quiver of $A$. As a corollary, we obtain a precise answer to a question posed by Chaparro, Schroll and Solotar.