Representations in infinite-dimensional lie theory

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Lie algebras are mathematical objects that describe the symmetries of certain physical systems.  Since many such systems have infinitely many independent symmetries, it is natural to study infinite-dimensional Lie algebras.  For example, one of the most important families of infinite-dimensional Lie algebras, called the affine Lie algebras, come from symmetries occurring in string theory.Even more important than the Lie algebras themselves are the ways in which they act as symmetries of spaces, called representations.  A Lie algebra typically has a great many representations, and understanding them can give more information about the Lie algebra, or even about the physics of the system from which the Lie algebra arises.The goal of this research program is to investigate the structure and representations of generalizations of the affine Lie algebras, with special emphasis on multivariable analogues of these Lie algebras.  In particular, we expect to give the first description and classification of several such representations, and we will use internal symmetries (called automorphisms) to draw conclusions about the structure of these objects.