Lie superalgebra modules of constant Jordan type

TL;DR

This paper extends the theory of modules of constant Jordan type to Lie superalgebras, introducing new concepts, properties, and classifications, including the study of endotrivial modules and super vector bundles over projective space.

Contribution

It develops the theory of constant super Jordan type modules for Lie superalgebras, providing definitions, properties, and classifications, especially for the detecting subalgebra _1 = (1|1).

Findings

Defined and analyzed modules of constant super Jordan type.

Constructed super vector bundles over projective space.

Classified modules of constant super Jordan type for (1|1).

Abstract

The theories of $\pi$-points and modules of constant Jordan type have been a topic of much recent interest in the field of finite group scheme representation theory. These theories allow for a finite group scheme module $M$ to be restricted down and considered as a module over a space of small subgroups whose representation theory is completely understood, but still provide powerful global information about the original representation of $M$. This paper provides an extension of these ideas and techniques to study finite dimensional supermodules over a classical Lie superalgebra $\mathfrak{g} = \mathfrak{g}_{\overline{0}} \oplus \mathfrak{g}_{\overline{1}}$. Definitions and examples of $\mathfrak{g}$-modules of constant super Jordan type are given along with proofs of some properties of these modules. Additionally, endotrivial modules (a specific case of modules of constant Jordan type) are studied. The case when $\mathfrak{g}$ is a detecting subalgebra, denoted $\mathfrak{f}_r$, of a stable Lie superalgebra is considered in detail and used to construct super vector bundles over projective space $\mathbb{P}^{r-1}$. Finally, a complete classification of supermodules of constant super Jordan type are given for $\mathfrak{f}_1 = \mathfrak{sl}(1|1)$.