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Spectral extension of the quantum group cotangent bundle
Communications in Mathematical Physics. 2009. Vol. 288. P. 1137-1179.
Isaev A. P., Pyatov P. N.
The structure of a cotangent bundle is investigated for quantum linear groups
GLq (n) and SLq (n). Using a q-version of the Cayley-Hamilton theorem we construct
an extension of the algebra of differential operators on SLq (n) (otherwise called the
Heisenberg double) 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—theWeyl partners of the spectral
variables. Calculating defining relations in terms of these generators allows us to
derive SLq (n) type dynamical R-matrices in a surprisingly simple way. Second, we
calculate an evolution operator for the model of the 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. The relation between the two operators is given by a modular functional equation for the Riemann theta function.
Research target:
Philosophy, Ethics, and Religious Studies
Language:
English