Representations of Virasoro and Affine Lie Algebras

Abstract

The first important results in the theory of representations of infinite-dimensional Lie groups were obtained in the book by Friedrichs “Mathematical problems of quantum field theory” (1953), which inspired whole series of papers on automorphisms of the commutation relations. A summary of this evolution was given in the book by F.A.Berezin “Methods of second quantization” (Berezin (1965)). The basic result cotained therein was the description of explicit formulas for spinor representations of the group (O(2∞, ℝ), U(∞)) and for the Weil representation of the group (Sp(2∞, ℝ), U(∞)), where the symbols (O(2∞, ℝ), U(∞)) (resp. (Sp(2∞, ℝ), U(∞))) denote the group of all orthogonal (resp. symplectic) operators that can be written as a sum of a unitary and a Hilbert-Schmidt operator (for more details see Sect.3).