Representation theory and quantum integrability

2004/02/07 by A. Gerasimov, Gerasimov, A., S. Kharchev +3
Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Algebra (math.QA) #Representation Theory (math.RT) #math-ph #math.MP #math.QA #math.RT

paper · pdf · doi:10.48550/arxiv.math/0402112

AmsLaTex, 24 pages; Section 3 is revised

arxiv created 2004/03/28 · arxiv updated 2009/12/01

Abstract

We describe new constructions of the infinite-dimensional representations of U(\mathfrakg) and U_q(\mathfrakg) for \mathfrakg being \mathfrakgl(N) and \mathfraksl(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 direct connection between group theoretical approach to the quantum integrability and Quantum Inverse Scattering Method based on the representation theory of Yangian and its generalizations. In the case of U_q(\mathfrakg) the considered representation is naturally supplied with the structure of U_q(\mathfrakg)⊗ U _q(\check\mathfrakg)-bimodule where \check \mathfrakg is Langlands dual to \mathfrakg and log q/2πi=- (log q/2πi)^-1. This bimodule structure is a manifestation of the Morita equivalence of the algebra and its dual.

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