Khazret S. Nirov

Exercises with the universal R-matrix

Using the formula for the universal R-matrix proposed by Khoroshkin and Tolstoy, we give a detailed derivation of L-operators for the quantum groups associated with the generalized Cartan matrices A(1)1 and A(1)2.

Universal

We collect and systematize general definitions and facts on the application of quantum groups to the construction of functional relations in the theory of integrable systems. As an example, we reconsider the case of the quantum group related to the six-vertex model. We prove the full set of the functional relations in the form independent of the representation of the quantum group in the quantum space and specialize them to the case of the six-vertex model.

R

A detailed construction of the universal integrability objects related to the integrable systems associated with the quantum group

-matrix and functional relations

\mathrm U_q(\mathcal L(\mathfrak{sl}_2))

Quantum groups and functional relations for higher rank

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

Oscillator versus prefundamental representations

\mathrm U_q(\mathcal L(\mathfrak{sl}_3))

W-algebras for non-abelian Toda systems

with an extensions to the higher spin case.

BRST FORMALISM FOR SYSTEMS WITH HIGHER ORDER DERIVATIVES OF GAUGE PARAMETERS

For the case of quantum loop algebras

Quantum groups, Verma modules and

\mathrm U_q(\mathcal L(\mathfrak{sl}_{l + 1}))

q

with