Certain Weyl Modules of Infinite Dimensional Lie Superalgebras

Abstract

The notion of a Weyl module for classical affine algebras, a type of infinite dimensional Lie algebra, was introduced in 2001 by Chari and Pressley. These modules are universal, finite dimensional highest weight modules. We expand these ideas to infinite-dimensional Lie superalgebras; in particular to Lie superalgebras of the form $\mathfrak{g} \otimes A$, where (\mathfrak{g}) is isomorphic to one of the superalgebras $\mathfrak{sl}_2(\mathbb{C})$, $\mathfrak{sl}(1 , | , 1)$, and $\mathfrak{osp}(1 , | 2 ) $ and $A$ is $\mathbb{C}[t]$ or $\mathbb{C}[t, t^{-1}]$. We prove that these Weyl modules are universal, finite-dimensional, highest weight $\mathfrak{g} \otimes A $-modules. We find that the weights satisfy a recurrence relation, and in the case of $\mathfrak{g} = \mathfrak{sl}(1 , | , 1)$, provide a formula for the dimension of each weight space. We also provide irreducibility criteria and the structure of the composition series. We then study $ \mathfrak{sl}(1 , | , 1) \otimes A$ modules generated by an infinite set of highest weight vectors, and with finite support. For these modules, we provide irreducibility criteria, find that the weights are of exponential-polynomial form, and provide a formula for the dimension of each weight space.