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Discrete evolution operator for $q$-deformed isotropic top

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Discrete Integrable Systems

The structure of a cotangent bundle is investigated for quantum linear groups $GL_q(n)$ and $SL_q(n)$. Using a $q$version of the Cayley-Hamilton theorem we construct an extension of the algebra of differential operators on $SL_q(n)$ by spectral values of the matrix of right invariant vector fields. We consider two applications for the spectral extension. First, we describe the extended Heisenberg double in terms of a new set of generators— the Weyl partners of the spectral variables. Calculating defining relations in terms of these generators allows us to derive $SL_q(n)$ type dynamical R-matrices in a surprisingly simple way. Second, we calculate discrete evolution operator for the model of $q$-deformed isotropic top introduced by A.Alekseev and L.Faddeev. The evolution operator is not uniquely defined and we present two possible expressions for it. The first one is a Riemann theta function in the spectral variables. The second one is an almost free motion evolution operator in terms of logarithms of the spectral variables. Relation between the two operators is given by a modular functional equation for Riemann theta function.

This talk is part of the Isaac Newton Institute Seminar Series series.

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