FRENKEL, 97-00765 Igor Frenkel plans to continue his work on representation theory of affine Lie algebras, and related algebraic structures such as vertex operator algebras and quantum groups. The author proposes several specific problems that should contribute to the development of these subjects and to better understanding of their interrelations. Frenkel also plans to study representation theory of a new class of Lie algebras and groups, recently introduced by him and his collaborators. Besides its intrinsic mathematical significance the new theory has important applications to four-dimensional quantum field theory in physics. Representation theory of Lie algebras and Lie groups initially developed at the turn of the century studies the structure of symmetries and their realizations. This theory encompasses many areas in mathematics and has fundamental applicaitons to theoretical physics. In the past twenty years the study of representaiton theory of a special class of affine Lie algebras and Lie groups led to many new unexpected discoveries in mathematics and theoretical physics as well as to a synthesis of many established areas in both disciplines. In particular, it yielded a pure mathe- matical description of a simplest quantum field theory -- a prototype theory of fundamental interactions. It is expected that representation theory of new classes of infinite-dimensional Lie algebras and groups might substantially deepen our understanding of mathematics and theoretical physics during the next decade.
