Various constructions of the affine Lie algebra action on the moduli space of instantons on 4-manifolds are discussed. The analogy between the local-global principle and the role of mass is also explained. The detailed proofs are given in separated papers [16, 17].

Given a simple finite-dimensional Lie algebra and an automorphism of finite order, one defines the notion twisted toroidal algebra. In this paper, we construct representations algebras from modules over affine lattice vertex algebras.

Generalizing the characters of compact simple Lie groups, Ian Macdonald introduced in [M1,M2] and other works remarkable orthogonal symmetric polynomials dependent on the parameters q, t. He came up with three main conjectures formulated for arbitrary root systems. A new approach to the Macdonald theory was suggested in [C1] on the basis of (double) affine Hecke algebras and related difference ...

Rogers-Ramanujan type partition identities establish equalities between certain types of partitions with difference conditions and partitions whose generating functions is an infinite product. Since the 1980’s, many connections between representations of Lie algebras and RogersRamanujan type partition identities have emerged. Lepowsky and Wilson [6] were the first to establish this link by givi...