The first Hochschild cohomology as a Lie algebra

In this paper we study sufficient conditions for the solvability of first Hochschild cohomology a finite dimensional algebra as Lie in terms its Ext-quiver arbitrary characteristic. particular, show that if quiver has no parallel arrows and loops then is solvable. For quivers containing loops, determine easily verifiable cohomology. We apply these criteria to space large families algebras, namely, several self-injective tame algebras including all blocks groups some wild most quantum complete intersections.