Supersymmetric Picard-Vessiot Theory, I: Basic Theory

TL;DR

This paper extends Picard-Vessiot theory to supersymmetric contexts using Hopf-algebraic methods, defining SUSY fields and generalizing PV extensions with superalgebraic structures.

Contribution

It introduces a supersymmetric version of PV theory, defining SUSY fields and extending Hopf-Galois extensions to the superalgebraic setting.

Findings

Defines SUSY fields and their properties

Generalizes PV extensions to superalgebraic context

Highlights the role of Hopf-Galois extensions in SUSY PV theory

Abstract

M. Takeuchi (1989) proposed a Hopf-algebraic approach to Picard-Vessiot (or PV) theory, giving a new definition of PV extensions by which such extensions become more smoothly connected, through Hopf-Galois extensions, to the associated affine group schemes. This paper extends PV theory to the supersymmetric (or SUSY) context, following Takeuchi's approach. The notion of SUSY fields is defined. Differential fields in the existing PV theory are replaced by $D$-SUSY fields, where $D$ is an acting super-cocommutative Hopf superalgebra. Hopf-Galis extensions here play a more and more crucial role.