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

For an arbitrary finite-dimensional algebra $A$, we first provide a sufficient condition for the solvability of its first Hochschild cohomology, considered as a Lie algebra, in terms of a corresponding separated quiver. If $A$ is moreover of non-wild representation type, we then determine a Levi subalgebra of this Lie algebra. It turns out always to be a sum of copies of $\mathfrak{sl}_2$, the precise number of which we give an explicit formula for. The formula is in terms of certain chains of Kronecker subquivers of $A$, which can easily be counted. In particular, whether the first Hochschild cohomology of a non-wild algebra is solvable can readily be read off from a presentation. This answers a question posed by Chaparro, Schroll and Solotar.