Representations in Infinite Dimensional Lie Theory

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Lie algebras are mathematical structures that describe symmetries of certain physical systems. Since many such systems have infinitely many independent symmetries, it is important to study infinite dimensional Lie algebras. For example, some of the most interesting infinite dimensional Lie algebras, called affine Kac-Moody algebras, describe symmetries in string theory. Even more important than the Lie algebras themselves are the ways in which they act as symmetries of geometric spaces, called representations. A Lie algebra typically has a great many representations, and understanding them gives information about the ambient geometric and physical context. We are particularly interested in families of Lie algebras that include, but also go beyond, the affine Kac-Moody algebras. Algebras like these help us better understand the geometry around affine Kac-Moody algebras, as well as give hints of more complex higher dimensional structures. Our work is expected to identify all the Lie algebras with certain useful properties, to provide a common framework for studying the representations of two different families of Lie algebras, and to develop new techniques for understanding infinite dimensional geometry related to particle physics.