Infinite-Dimensional Lie (Super) Algebras and Related Algebraic Structures

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9970007The main part of this proposal deals with the algebraic structure, called a conformal algebra, which is an axiomatic description of the operator product expansion of chiral fields in conformal field theory. The main tool used in the structure theory of conformal algebras is the Cartan-Lie theory of analytic pseudogroups. Thus, in order to develop a structure theory of conformal superalgebras one needs an extension of the Cartan-Lie theory to the super case. The classification of simple analytic super pseudogroups quite unexpectedly turned out to be very rich and beautiful. These new objects require a careful study. The work is also underway on a structure theory of multidimensional conformal algebras. Other items of the proposal include representation theory of W_infinity type of Lie algebras, a character theory for affine Lie superalgebras and its connections to number theory, and the theory of quantum orbifolds.There are deep reasons to believe that the new theory of simple super pseudogroups should play a fundamental role in further understanding of the Standard model, the quantum theory of electromagnetic, strong and weak interactions. The theory of conformal algebras is deeply related to the theory of infinite-dimensinal Lie algebras and superalgebras and may lead to a better understanding of the operator product expansion in realistic quantum field theories. The character theory of affine Lie superalgebras is intimately connected to some recent developments in the theory of modular forms.