Supersymmetric Picard-Vessiot Theory, I: Basic Theory

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.