Certain Weyl Modules of Infinite Dimensional Lie Superalgebras
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Authors
Ruzic, Sonja
Date
2024-05-28
Type
thesis
Language
eng
Keyword
mathematics , lie superalgebras , representation theory
Alternative Title
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.
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Copying and Preserving Your Thesis
This publication is made available by the authority of the copyright owner solely for the purpose of private study and research and may not be copied or reproduced except as permitted by the copyright laws without written authority from the copyright owner.
Queen's University's Thesis/Dissertation Non-Exclusive License for Deposit to QSpace and Library and Archives Canada
ProQuest PhD and Master's Theses International Dissemination Agreement
Intellectual Property Guidelines at Queen's University
Copying and Preserving Your Thesis
This publication is made available by the authority of the copyright owner solely for the purpose of private study and research and may not be copied or reproduced except as permitted by the copyright laws without written authority from the copyright owner.