SciPost Phys. 12, 146 (2022) ·
published 4 May 2022
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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.
A. Hutsalyuk, A. Liashyk, S. Z. Pakuliak, E. Ragoucy, N. A. Slavnov
SciPost Phys. 4, 006 (2018) ·
published 30 January 2018
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We obtain recursion formulas for the Bethe vectors of models with periodic
boundary conditions solvable by the nested algebraic Bethe ansatz and based on
the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_{n})$. We also present
a sum formula for their scalar products. This formula describes the scalar
product in terms of a sum over partitions of the Bethe parameters, whose
factors are characterized by two highest coefficients. We provide different
recursions for these highest coefficients. In addition, we show that when the
Bethe vectors are on-shell, their norm takes the form of a Gaudin determinant.
