Vertex Operator Algebras & Lie Algebras

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9401851 Lepowsky The principal investigators will investigate vertex operator algebras and Lie algebras. One will study the new concept of vertex tensor category. The other will construct a new type of basis for the free Lie algebra and to study affinization of certain simple Lie algebras of Cartan type in prime characteristic. Together they will use vertex operator algebra theory to study Z-algebras. Conformal field theory is an important physical theory describing both two-dimensional critical phenomena in condensed matter physics and classical motions of strings in string theory. Besides its importance in physics, the beautiful and rich mathematical structure of conformal field theory also has interested many mathematicians. New relations between different branches of mathematics, such as representations of infinite-dimensional Lie algebras and groups, Riemann surfaces and algebraic curves, the Monster sporadic group, modular functions and modular forms, elliptic genera, and knot theory, is revealed in the study of conformal field theory.