Algebraic groups as automorphism groups of algebras

2020/12/10 by J. S. Milne, Milne, James S
Mathematics · #14L15 14C30 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2012.05708

openalex publication_date 2020/12/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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