Almost nilpotency of an associative algebra with an almost nilpotent fixed-point subalgebra

TL;DR

This paper proves that an associative algebra with a finite soluble automorphism group, which has a fixed-point subalgebra containing a large nilpotent ideal, must itself contain a large nilpotent ideal, with bounds depending on the group and subalgebra properties.

Contribution

It establishes a link between the nilpotency properties of a fixed-point subalgebra and the entire algebra under automorphisms of finite soluble groups.

Findings

Existence of a nilpotent ideal in A with bounded index

Bounded codimension of the nilpotent ideal in A

Dependence of bounds on group order, nilpotency index, and codimension

Abstract

Let A be an associative algebra of arbitrary dimension over a field F and G a finite soluble group of automorphisms of A oforder n, prime to the characteristic of F. We prove that if the fixed-point subalgebra of A under the action of G contains a two-sided nilpotent ideal of nilpotency index d and of finite codimension m in I, then A contains a nilpotent two-sided ideal of nilpotency index bounded by a function of n and d and of finite codimension bounded by a function of m, n and d.