Solvability and nilpotency for algebraic supergroups

TL;DR

This paper investigates the properties of algebraic supergroups over fields with characteristic not equal to 2, establishing conditions for solvability and nilpotency, and extending classical theorems to the supergroup context.

Contribution

It provides new criteria for solvability and nilpotency of algebraic supergroups, including a super-analogue of the Chevalley Decomposition Theorem, and characterizes smooth superalgebras.

Findings

Algebraic supergroups are solvable if their even part is trigonalizable.

Characterization of nilpotent connected algebraic supergroups.

A super-analogue of the Chevalley Decomposition Theorem is established.

Abstract

We study solvability, nilpotency and splitting property for algebraic supergroups over an arbitrary field $K$ of characteristic $\mathrm{char}\, K \ne 2$. Our first main theorem tells us that an algebraic supergroup $\mathbb{G}$ is solvable if the associated algebraic group $\mathbb{G}_{ev}$ is trigonalizable. To prove it we determine the algebraic supergroups $\mathbb{G}$ such that $\dim \mathrm{Lie}(\mathbb{G})_1=1$; their representations are studied when $\mathbb{G}_{ev}$ is diagonalizable. The second main theorem characterizes nilpotent connected algebraic supergroups. A super-analogue of the Chevalley Decomposition Theorem is proved, though it must be in a weak form. An appendix is given to characterize smooth Noetherian superalgebras as well as smooth Hopf superalgebras.