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

Eisele, F. ORCID: 0000-0001-8267-2094 & Raedschelders, T. (2020). On solvability of the first Hochschild cohomology of a finite-dimensional algebra. Transactions of the American Mathematical Society, doi: 10.1090/tran/8064

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.