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

For an arbitrary finite-dimensional algebra A A , we introduce a general approach to determining when its first Hochschild cohomology H H 1 ( A ) \mathrm {HH}^1(A) , considered as a Lie algebra, is solvable. If A A is, moreover, of tame or finite representation type, we are able to describe H H 1 ( A ) \mathrm {HH}^1(A) as the direct sum of a solvable Lie algebra and a sum of copies of s l 2 \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 A . As a corollary, we obtain a precise answer to a question posed by Chaparro, Schroll, and Solotar.