Automorphism group schemes and Weyl groups of gradings

TL;DR

This paper extends the classical understanding of automorphism groups of algebraic gradings to a scheme-theoretic framework, revealing the structure of Weyl group schemes over arbitrary fields.

Contribution

It introduces a scheme-theoretic approach to automorphism groups of gradings and generalizes the Weyl group concept beyond algebraically closed fields.

Findings

Automorphism group of a grading is the normalizer of its diagonal group over any field.

The quotient of automorphism group and stabilizer forms a constant group scheme called the Weyl group scheme.

Over arbitrary fields, the Weyl group scheme differs from the classical Weyl group associated with algebraically closed fields.

Abstract

A scheme theoretic version of the automorphism group of a grading on an algebra is presented, and the classical result that shows that, over algebraically closed fields of characteristic 0, the automorphism group of a grading is the normalizer of its diagonal group is extended, over arbitrary fields, to this scheme setting. The quotient of the scheme theoretic versions of the automorphism group and the stabilizer of a grading turns out to be a constant group scheme, called the Weyl group scheme of the grading. For algebraically closed fields this is the constant group scheme associated to the ordinary Weyl group of the grading, but this fails over arbitrary fields.