Algebraic supergroups and Harish-Chandra pairs over a commutative ring

TL;DR

This paper establishes an equivalence between algebraic supergroups and Harish-Chandra pairs over certain rings, and applies this to reconstruct known supergroups and describe their representations via super-hyperalgebras.

Contribution

It proves a category equivalence over 2-torsion free rings and reconstructs Chevalley supergroups, advancing the understanding of supergroup structures and their representations.

Findings

Proves category equivalence between algebraic supergroups and Harish-Chandra pairs

Reconstructs Chevalley $\\mathbb{Z}$-supergroups using the equivalence

Describes representations of supergroups via super-hyperalgebras

Abstract

We prove a category equivalence between algebraic supergroups and Harish-Chandra pairs over a commutative ring which is $2$-torsion free. The result is applied to re-construct the Chevalley $\mathbb{Z}$-supergroups constructed by Fioresi and Gavarini [8] and by Gavarini [9, 10]. For a wide class of algebraic supergroups we describe their representations by using their super-hyperalgebras.