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Baecklund transformations and Baxter's Q-operator
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The course of 5 lectures given at the seminar "Integrable Systems: from Classical to Quantum" (Universite de Montreal, Jul 26 -- Aug 6, 1999) contains a detailed comment on the recently discovered (Gaudin-Pasquier, 1992) connection between Baecklund transformations in the theory of classical integrable systems on one hand, and Baxter's Q-operator for quantum integrable systems, on the other hand. We restrict our attention to the systems with finite number of degrees of freedom. Our main illustrative example is the periodic Toda lattice. We present a general construction of Q-operator for models governed by the SL(2)-invariant R-matrix and apply it to our example. We discuss also applications of BT and Q-operators to the separation of variables and theory of special functions.
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Nonabelian shift operators and shifted Yangians
New nonabelian shift operators and wall-crossing identities show that the quantized Coulomb branch of pure GL_n gauge theory is a quotient of the shifted Yangian Y_{-nα}(sl2), with vertex functions as Hecke eigenfunctions.
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