Spectral extension of the quantum group cotangent bundle

TL;DR

This paper extends the algebraic structure of quantum groups by incorporating spectral values, leading to new generators and simplified derivation of dynamical R-matrices, with applications to q-deformed tops.

Contribution

It introduces a spectral extension of the quantum group cotangent bundle, providing new generators and methods for deriving R-matrices and evolution operators.

Findings

Extended Heisenberg double with spectral variables and Weyl partners.

Derived SLq(n) type dynamical R-matrices using spectral generators.

Presented two forms of the evolution operator for q-deformed isotropic top.

Abstract

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 -- the Weyl 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 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.