A new equivalence between super Harish-Chandra pairs and Lie supergroups

Fabio Gavarini

Vol. 306 (2020), No. 2, 451–485

DOI: 10.2140/pjm.2020.306.451

Abstract

It is known that there exists a natural functor $\Phi$ from Lie supergroups to super Harish-Chandra pairs. A functor going backwards, that associates a Lie supergroup with each super Harish-Chandra pair, yielding an equivalence of categories, was found by Koszul (1983), and later generalized by several authors.

We provide two new backwards equivalences, i.e., two different functors ${\Psi }^{\circ }$ and ${\Psi }^e$ that construct a Lie supergroup (thought of as a special group-valued functor) out of a given super Harish-Chandra pair, so that both ${\Psi }^{\circ }$ and ${\Psi }^e$ are quasi-inverse to the functor $\Phi$.

Keywords

Lie supergroups, super Harish-Chandra pairs, Lie superalgebras

Mathematical Subject Classification

Primary: 14M30, 58A50

Secondary: 17B99

Milestones

Received: 3 December 2019

Revised: 7 May 2020

Accepted: 7 May 2020

Published: 13 July 2020