Generalizations of the Lie superalgebras of supermatrices of complex size and related topics

Abstract

A class of simple filtered Lie algebras of polynomial growth with increasing filtration is distinguished and presentations of these algebras are explicitely described for the simplest examples. Lie (super)algebras of this class appear in relation with Calogero--Sutherland model, high-spin supergravity, etc.; they are associated with the associative algebras of twisted differential operators on the big Schubert cell of the flag varieties. The Lie algebra of matrices of complex size introduced by Feigin is a simplest example of our algebras. Usually, they posess a trace and an invariant symmetric bilinear form; hence, analogs of dynamical systems such as Yang-Baxter, KdV, Leznov--Saveliev, etc. are associated with them. In particular, in the space of pseudodifferential operators there are analogs of the KdV hierarchies associated with sl(n) for n complex in the same way as the KdV hierarchy is associated with sl(n) for n integer are those studied by Gelfand--Dickey and Khesin--Malikov. We briefly describe such dynamical systems, and generalizations of the classical orthogonal polynomials.