On the R-matrix realization of quantum loop algebras

TL;DR

This paper develops an R-matrix realization of quantum loop algebras for non-exceptional affine Lie types, providing new relations among Gauss coordinates and currents crucial for quantum integrable models.

Contribution

It introduces a novel R-matrix realization of quantum loop algebras and derives explicit relations between Gauss coordinates and currents for various affine types.

Findings

Derived commutation relations for Gauss coordinates.

Presented a new realization of quantum loop algebras in terms of currents.

Established relations between off-diagonal Gauss coordinates and current projections.

Abstract

We consider $\rm R$-matrix realization of the quantum deformations of the loop algebras $\tilde{\mathfrak{g}}$ corresponding to non-exceptional affine Lie algebras of type $\hat{\mathfrak{g}}=A^{(1)}_{N-1}$, $B^{(1)}_n$, $C^{(1)}_n$, $D^{(1)}_n$, $A^{(2)}_{N-1}$. For each $U_q(\tilde{\mathfrak{g}})$ we investigate the commutation relations between Gauss coordinates of the fundamental $\mathbb{L}$-operators using embedding of the smaller algebra into bigger one. The new realization of these algebras in terms of the currents is given. The relations between all off-diagonal Gauss coordinates and certain projections from the ordered products of the currents are presented. These relations are important in applications to the quantum integrable models.