Linear supergroup actions. I. On the defining properties

Abstract:

This paper studies the notions of linearity and bilinearity in the category of supermanifolds. Following the work begun by [OASV2], we deal with supermanifoldifications of supervector spaces. The ${{\mathbf {R}}^{1|1}}$-module operations are defined componentwise. The linearity and bilinearity properties are stated by requiring commutativity of some appropriate diagrams of supermanifold morphisms. It is proved that both linear and bilinear supermanifold morphisms are completely determined by their underlying continuous maps, which in turn have to be linear (resp., bilinear) in the usual sense. It is observed that whereas linear supermanifold morphisms are vector bundle maps, bilinear supermanifold morphisms are not. A natural generalization of the bilinear evaluation map $\operatorname {Hom} (V, W) \times V \to W\;((F, v) \mapsto F(v))$ is given and some applications pointing toward the notions of linear supergroup actions and adjoint and coadjoint actions are briefly discussed.