Date of Award
May 2015
Degree Type
Dissertation
Degree Name
Doctor of Philosophy
Department
Mathematics
First Advisor
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.
Recommended Citation
Schleben, Bradford J., "Infinitely Generated Clifford Algebras and Wedge Representations of gl∞|∞" (2015). Theses and Dissertations. 919.
https://dc.uwm.edu/etd/919