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

TL;DR

This paper proves Halperin's conjecture that the automorphism group of a finite-dimensional algebra with certain properties is solvable, completing previous partial results and analyzing extremal cases to simplify proofs of related theorems.

Contribution

It completes the proof of Halperin's conjecture on the solvability of automorphism groups for certain algebras and refines criteria for non-solvable derivation algebras.

Findings

Halperin's conjecture is fully proved.

Extremal cases where derivation algebra is non-solvable are characterized.

A simplified, self-contained proof of Yau's theorem is provided.

Abstract

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