On the R-matrix realization of quantum loop algebras

Abstract

We consider rr-matrix realization of the quantum deformations of the loop algebras \tilde{\mathfrak{g}}𝔤̃ corresponding to non-exceptional affine Lie algebras of type \widehat{\mathfrak{g}}=A^{(1)}_{N-1}𝔤̂=AN−1(1), B^{(1)}_nBn(1), C^{(1)}_nCn(1), D^{(1)}_nDn(1), A^{(2)}_{N-1}AN−1(2). For each U_q(\tilde{\mathfrak{g}})Uq(𝔤̃) we investigate the commutation relations between Gauss coordinates of the fundamental \mathbb{L}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.