On functor points of affine supergroups

TL;DR

This paper simplifies the construction of affine supergroups from Harish-Chandra pairs using Hopf superalgebras, and applies it to describe normalizers, centralizers, and tensor product decompositions.

Contribution

It provides a more conceptual and generalized approach to constructing affine supergroups, improving upon Gavarini's method, and clarifies the differences in dualities via cocycle deformation.

Findings

Explicit description of normalizers and centralizers in supergroups

A tensor product decomposition theorem for Hopf superalgebras

Clarification of duality differences through cocycle deformation

Abstract

To construct an affine supergroup from a Harish-Chandra pair, Gavarini [2] invented a natural method, which first constructs a group functor and then proves that it is representable. We give a simpler and more conceptual presentation of his construction in a generalized situation, using Hopf superalgebras over a superalgebra. As an application of the construction, given a closed super-subgroup of an algebraic supergroup, we describe the normalizer and the centralizer, using Harish-Chandra pairs. We also prove a tensor product decomposition theorem for Hopf superalgebras, and describe explicitly by cocycle deformation, the difference which results from the two choices of dualities found in literature.