Representation theory and quantum integrability

Details Representation theory and quantum integrability Representation theory and quantum integrability

2004-02-07

arxiv.org/abs/math/0402112v2

Mathematics

Quantum Algebra

AmsLaTex, 24 pages; Section 3 is revised

Scientific paper

We describe new constructions of the infinite-dimensional representations of $U(\mathfrak{g})$ and $U_q(\mathfrak{g})$ for $\mathfrak{g}$ being $\mathfrak{gl}(N)$ and $\mathfrak{sl}(N)$. 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 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 $U_q(\mathfrak{g})$ the considered representation is naturally supplied with the structure of $U_q(\mathfrak{g})\otimes U_{\tilde q}(\check{\mathfrak{g}})$-bimodule where $\check {\mathfrak{g}}$ is Langlands dual to $\mathfrak{g}$ and $\log q/2\pi i=- (\log{\tilde q}/2\pi i)^{-1}$. This bimodule structure is a manifestation of the Morita equivalence of the algebra and its dual.

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