Fine gradings and automorphism groups on associative algebras

TL;DR

This paper characterizes automorphisms and gradings of associative algebras, providing criteria for when gradings are fine and computing automorphism groups, especially over the real numbers.

Contribution

It establishes that inner automorphisms stabilizing graded-simple unital associative algebras are conjugations by homogeneous elements, and computes automorphism groups for graded-division algebras.

Findings

Inner automorphisms are conjugations by homogeneous elements.

Necessary and sufficient conditions for gradings to be fine.

Automorphism groups expressed via graded-division algebra automorphisms.

Abstract

First we prove that any inner automorphism in the stabilizer of a graded-simple unital associative algebra whose grading group is abelian is the conjugation by a homogeneous element. Now consider a grading by an abelian group on an associative algebra such that the algebra is graded-simple and satisfies the DCC on graded left ideals. We give necessary and sufficient conditions for the grading to be fine. Then we assume that one of these necessary conditions to be fine is satisfied, and we compute the automorphism groups of the grading; the results are expressed in terms of the automorphism groups of a graded-division algebra. Finally we compute the automorphism groups of graded-division algebras in the case in which the ground field is the field of real numbers, and the underlying algebra (disregarding the grading) is simple and of finite dimension.