Torsors in super-symmetry

TL;DR

This paper explores super-torsors under affine super-groups in super-symmetry, establishing a correspondence with ordinary torsors and advancing the understanding of super-symmetric Picard-Vessiot theory.

Contribution

It demonstrates that affine super-torsors correspond exactly to ordinary torsors, extending classical torsor theory into the super-symmetric context.

Findings

Super-torsors under affine super-groups are in one-to-one correspondence with ordinary torsors.

Explicit conditions under which super-torsors arise from ordinary torsors are provided.

Results support the development of super-symmetric Picard-Vessiot theory.

Abstract

Torsors under affine groups are generalized in the super context by super-torsors under affine super-groups. We investigate those super-torsors by using Hopf-algebra language and techniques. It is explicitly shown, under suitable assumptions, that every super-torsor arises from an ordinary torsor. Especially, the objects with affinity restriction, or namely, the affine super-torsors and the affine ordinary torsors are shown to be precisely in one-to-one correspondence. The results play substantial roles in ongoing construction of super-symmetric Picard-Vessiot theory.