Representation theory and quantum integrability

Abstract

We describe new constructions of the infinite-dimensional representations of U(g) and Uq(g) for g being gl (N) and sl(N). The application of these constructions to the quantum integrable theories of Toda type is discussed. With the help of these infinite-dimensional representations we manage to establish a direct connection between the group theoretical approach and the Quantum Inverse Scattering Method based on the representation theory of the Yangian and its generalizations. In the case of Uq(g) the considered representation is naturally supplied with the structure of a (formula presented)-bimodule where (formula presented) is the Langlands dual to g and log (formula presented). This bimodule structure is a manifestation of the Morita equivalence of the algebra and its dual.