Chevalley Supergroups

TL;DR

This paper constructs supergeometric analogues of Chevalley groups within algebraic supergeometry, providing a unified framework for classical Lie superalgebras and introducing methods to build new algebraic supergroups.

Contribution

It introduces a scheme-theoretic construction of Chevalley supergroups, covering both basic and strange cases, and establishes foundational theorems for Lie superalgebras and their superalgebras.

Findings

Constructed affine algebraic supergroups from simple Lie superalgebras.

Proved existence of Chevalley bases for classical Lie superalgebras.

Developed a PBW-like theorem for Kostant superalgebras.

Abstract

In the framework of algebraic supergeometry, we give a construction of the scheme-theoretic supergeometric analogue of Chevalley groups, namely affine algebraic supergroups associated to simple Lie superalgebras of classical type. In particular both basic (even exceptional) and strange cases are covered. This provides a unified approach to most of the algebraic supergroups considered so far in literature, and an effective method to construct new ones. As an intermediate step, we prove an existence theorem for Chevalley bases of simple classical Lie superalgebras and a PBW-like theorem for their associated Kostant superalgebras.