# On the R-matrix realization of quantum loop algebras

### Submission summary

 As Contributors: Stanislav Pakuliak Arxiv Link: https://arxiv.org/abs/2106.10666v1 (pdf) Date submitted: 2021-06-23 07:41 Submitted by: Pakuliak, Stanislav Submitted to: SciPost Mathematics Academic field: Mathematics Specialties: Mathematical Physics

### 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 $\widehat{\mathfrak{g}}=A^{(1)}_{N-1}$, $B^{(1)}_n$, $C^{(1)}_n$, $D^{(1)}_n$, $A^{(2)}_{N-1}$. 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.

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