Infinitely Generated Clifford Algebras and Wedge Representations of gl∞|∞

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.