Discrete groups and the integrability of quantum systems

Miki Wadati

Hermann Boos, Frank Göhmann, Andreas Klümper +2 more

A detailed construction of the universal integrability objects related to the integrable systems associated with the quantum group is given. The full proof of the functional relations in the form independent of the representation of the quantum group on the quantum space is presented. The case of the general gradation and general twisting is treated. The specialization of the universal functional relations to the case when the quantum space is the state space of a discrete spin chain is described.

R. Bullough, JUSSI TIMONEN

S A Pol'shin

We construct the systems of generalized coherent states for the discrete and continuous spectra of the hydrogen atom. These systems are expressed in elementary functions and are invariant under the SO (3, 2) (discrete spectrum) and SO (4, 1) (continuous spectrum) subgroups of the dynamical symmetry group SO (4, 2) of the hydrogen atom. Both systems of coherent states are particular cases of the kernel of integral operator which intertwines irreducible representations of the SO (4, 2) group.

Khazret S. Nirov, A. V. Razumov

A detailed construction of the universal integrability objects related to the integrable systems associated with the quantum group $\mathrm U_q(\mathcal L(\mathfrak{sl}_2))$ is given. The full proof of the functional relations in the form independent of the representation of the quantum group on the quantum space is presented. The case of the general gradation and general twisting is treated. The specialization of the universal functional relations to the case when the quantum space is the state space of a discrete spin chain is described. This is a degression of the corresponding consideration for the case of the quantum group $\mathrm U_q(\mathcal L(\mathfrak{sl}_3))$ with an extensions to the higher spin case.

Leon A. Takhtajan