Representations of Lie superalgebras with fusion flags

TL;DR

This paper investigates the structure of finite-dimensional representations of basic classical Lie superalgebras, especially $ ext{osp}(1,2n)$, demonstrating the existence of fusion flags and providing presentations for fusion products.

Contribution

It introduces the concept of fusion flags in the category of representations of Lie superalgebras and offers explicit generators and relations for fusion products of modules.

Findings

Certain objects admit a fusion flag structure.

Fusion products of irreducible modules, Weyl modules, and Demazure modules are characterized.

A presentation with generators and relations for these fusion products is established.

Abstract

We study the category of finite--dimensional representations for a basic classical Lie superalgebra $\Lg=\Lg_0\oplus \Lg_1$. For the ortho--symplectic Lie superalgebra $\Lg=\mathfrak{osp}(1,2n)$ we show that certain objects in that category admit a fusion flag, i.e. a sequence of graded $\Lg_0[t]$--modules such that the successive quotients are isomorphic to fusion products. Among these objects we find fusion products of finite--dimensional irreducible $\Lg$--modules, truncated Weyl modules and Demazure type modules. Moreover, we establish a presentation for these types of fusion products in terms of generators and relations of the enveloping algebra.