Lie Algebra Representations and Modular Form Identities

This thesis explores the relationship between Lie algebras and modular forms in two different ways. The first part discusses infinite-dimensional affine Lie algebras, characters of which can in fact be viewed as modular forms. The exposition draws primarily from the texts of Humphreys and Kac, and concludes with the generalization of the Weyl Character formula to affine Lie algebras, and its applications. Special cases like the Weyl denominator formula give expansions for classical modular forms as infinite products, including the Macdonald identities for products of the Dedekind eta function. The second part considers generalizations of the Rankin-Cohen bracket, a differential operator acting on pairs of modular forms to produce a third. In this second part, which will be published in a forthcoming paper coauthored with Shaul Zemel, we establish minimality results for such combinations involving three modular forms, and begin an investigation into analogous structures for combinations of four or more modular forms. The representation theory of SL_2(C) is a key tool in our work.