On the Faulkner construction for generalized Jordan superpairs

TL;DR

This paper extends the Faulkner construction to generalized Jordan superpairs, establishing a bijective link with faithful Lie superalgebra modules and analyzing automorphism groups, with applications to tensor product constructions.

Contribution

It adapts the Faulkner construction to superalgebras, providing a bijective correspondence and automorphism group analysis for generalized Jordan superpairs.

Findings

Established a bijective correspondence between generalized Jordan superpairs and Lie superalgebra modules.

Proved isomorphism of automorphism group schemes for associated objects.

Extended tensor product construction to superpair class with good bilinear forms.

Abstract

In this paper, the well-known Faulkner construction is revisited and adapted to include the super case, which gives a bijective correspondence between generalized Jordan (super)pairs and faithful Lie (super)algebra (super)modules, under certain constraints (bilinear forms with properties analogous to the ones of a Killing form are required, and only finite-dimensional objects are considered). We always assume that the base field has characteristic different from $2$. It is also proven that associated objects in this Faulkner correspondence have isomorphic automorphism group schemes. Finally, this correspondence will be used to transfer the construction of the tensor product to the class of generalized Jordan (super)pairs with "good" bilinear forms.