Integrality over fixed rings of automorphisms in a Lie nilpotent setting

TL;DR

This paper proves that Lie nilpotent algebras over characteristic zero fields are integral over fixed rings of automorphisms, with specific degrees depending on the automorphism group and algebra properties.

Contribution

It establishes the integrality degree of Lie nilpotent algebras over fixed rings under automorphisms, extending classical results to a noncommutative Lie nilpotent setting.

Findings

R is right integral over Fix(G) of degree n^k

Skew polynomial algebra R[w,d] is right integral over Fix(d)[w^n]

Results extend classical integrality to Lie nilpotent algebras

Abstract

Let R be a Lie nilpotent algebra of index k over a field K of characteristic zero. If G is an n-element subgroup of Aut(R) of the K-automorphisms, then we prove that R is right integral over Fix(G) of degree n^k. In the presence of a primitive n-th root of unity e in K, for a K-automorphism d in Aut(R) with d^n=id, we prove that the skew polynomial algebra R[w,d] is right integral of degree n^k over Fix(d)[w^n].