Lie supergroups vs. super Harish-Chandra pairs: a new equivalence

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 [18]; this result was later extended by other authors, to different levels of generality, but always elaborating on Koszul's original idea. In this paper, I 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 any Lie supergroup is recovered from its naturally associated super Harish-Chandra pair; more precisely, both ${\Psi }^{\circ }$ and ${\Psi }^e$ are quasi-inverse to the functor $\Phi$.

47 pages. The present version is a far-reaching improvement of the first one: roughly, we redouble the main result(s). Sections 2 and 3 are essentially unchanged (but for minimal misprints), while sections 4, 5 and 6 are modified (enriched) a lot; accordingly, section 7 is (slightly) adapted too. I also added *Keywords*: Lie supergroups - Lie superalgebras - super Harish-Chandra pairs

References in corpus (2)

• A Categorical Formulation of Superalgebra and Supergeometry
• A general construction of Weil functors

Cited by in corpus (2)

• Real forms of complex Lie superalgebras and supergroups
• Singular Degenerations of Lie Supergroups of Type $D(2,1;a)$