Algebraic groups as automorphism groups of algebras

TL;DR

This paper demonstrates that any algebraic group scheme over sufficiently large fields can be realized as an automorphism group of a nonassociative algebra, providing new insights into algebraic Lie algebras and moduli spaces.

Contribution

It extends previous results to include algebraic groups over discrete valuation rings and offers new characterizations of algebraic Lie algebras.

Findings

Realization of algebraic groups as automorphism groups of nonassociative algebras

Simplified descriptions of Mumford--Tate domains and Shimura varieties

Extension of results to algebraic groups over discrete valuation rings

Abstract

We show that every algebraic group scheme over a field with at least 8 elements can be realized as the group of automorphisms of a nonassociative algebra. This is only a modest improvement of the theorem of Gordeev and Popov (2003), but it allows us to give a new characterization of algebraic Lie algebras and to simplify the standard descriptions of Mumford--Tate domains and Shimura varieties as moduli spaces. Once the original argument of Gordeev and Popov has been rewritten in the language of schemes, we find that it also applies to algebraic groups over discrete valuation rings.