On solvability of the automorphism group of a finite-dimensional algebra

Consider the automorphism group of a finite-dimensional algebra. S. Halperin conjectured that the identity component of this group is solvable if the algebra is a complete intersection. The solvability criterion recently obtained by M. Schulze [12] provides a proof to a local case of this conjecture as well as giving an alternative proof of S.S.-T. Yau's theorem [15] based on a deep result due to G. Kempf. In this note we complete the proof of Halperin's conjecture and study the extremal cases in Schulze's criterion, where the Lie algebra of derivations is non-solvable. This allows us to deduce a direct, self-contained proof of Yau's theorem.