General solution of the Yang–Baxter equation with symmetry group $\mathrm {SL}(\mathit {n},\mathbb {C})$

Abstract

Abstract. The problem of constructing the R-matrix is considered in the case of an integrable spin chain with symmetry group SL(n, C). A fairly complete study of general R-matrices acting in the tensor product of two continuous series representations of SL(n, C) is presented. On this basis, R-matrices are constructed that act in the tensor product of Verma modules (which are infinite-dimensional representations of the Lie algebra sl(n)), and also R-matrices acting in the tensor product of finite-dimensional representations of the Lie algebra sl(n).

The R-matrix factorization, Q-operator, and variable separation in the case of the XXX spin chain with the SL(2, C) symmetry group

“…The operator R 12 , first, permutes the indices 1 and 2 in the product L 1 L 2 , which corresponds to interchanging z 1 z 2 , and, second, permutes the pairs of parameters u 1 , u 2 and v 1 , v 2 . The SL(2, C)-invariant solution of the RRL equations has the form [13], [17] …”

“…We have thus derived key formulas (13) and (12) of the variable separation method using only the master equation for the main building block, the operator R 2 .…”

Matrix Factorization for Solutions of the Yang–Baxter Equation

“…This observation enabled to construct the general solution of the Yang-Baxter equation,Eq. (1), acting on a tensor product of two infinite-dimensional principal series representations of the group SL(N, C) [12]. In the case of rank one algebra this result has been carried over to trigonometric and elliptic deformations.…”

“…The equality of R-operators (4) and (5) (up to an inessential normalization factor), provided the functional realization of sℓ 2 , Eq. (3), is adopted, can be checked by a straightforward calculation [12]. The solution (5) of the Yang-Baxter equation has been constructed in [10] for infinite-dimensional representations of Verma module type.…”

“…The L-operator can be factorized in a product of several more simple matrices in the case of sℓ 2 symmetry algebra as well as in the case of its trigonometric and elliptic deformations [3,11,25]. This observation helps a lot in solving the RLL-relation, which imposes severe constraints on the infinitedimensional R-operator [7,[11][12][13] and eventually enables to find the general solution of the Yang-Baxter equation, Eq. (1).…”