Theses and Dissertations

May 2015

Dissertation

Degree Name

Doctor of Philosophy

Mathematics

Yi Ming Zou

Committee Members

Allen Bell, Craig Guilbault, Ian Musson, Jeb Willenbring

Keywords

Algebra, Clifford, Infinite Dimensional, Lie, Representation, Superalgebra

Abstract

The goal of this dissertation is to explore representations of $\mathfrak{gl}_{\infty|\infty}$ and associated Clifford superalgebras. The machinery utilized is motivated by developing an alternate superalgebra analogue to the Lie algebra theory developed by Kac. In an effort to establish a natural mathematical analogue, we construct a theory distinct from the super analogue developed by Kac and van de Leur. We first construct an irreducible representation of a Lie superalgebra on an infinite-dimensional wedge space that permits the presence of infinitely many odd parity vectors. We then develop a new Clifford superalgebra, whose structure is also examined. From here, we extend our representation to the central extension of this Lie superalgebra and develop a correspondence between a subsuperalgebra of that extension and the Clifford superalgebra previously constructed. Finally, we begin to provide a context to study all Clifford algebras of an infinite-dimensional non-degenerate real quadratic space.

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