Yang–Baxter -operators and parameter permutations

Abstract

We present an uniform construction of the solution to the Yang-Baxter equation with the symmetry algebra sℓ(2) and its deformations: the q-deformation and the elliptic deformation or Sklyanin algebra. The R-operator acting in the tensor product of two representations of the symmetry algebra with arbitrary spins ℓ 1 and ℓ 2 is built in terms of products of three basic operators S 1 , S 2 , S 3 which are constructed explicitly. They have the simple meaning of representing elementary permutations of the symmetric group S 4 , the permutation group of the four parameters entering the RLL-relation.1

Correlators with sℓ 2 Yangian symmetry

“…Therefore we can omit the superscripts ± in the ratio-coordinate form. The uniformization of both results is achieved also in the notation used in [26]. We substitute for the signature + case v = u + ℓ, but for the signature − case v = u − ℓ − 1.…”

“…The YB relation for S q12 can be checked directly using the factorized form (4.3) [26], see Appendix C.…”

Matrix Factorization for Solutions of the Yang–Baxter Equation

“…(5) is factorized. The origin and the meaning of this and other similar factorizations has been clarified in [11]. The equality of R-operators (4) and (5) (up to an inessential normalization factor), provided the functional realization of sℓ 2 , Eq.…”

“…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).…”

The hyperbolic modular double and the Yang-Baxter equation

“…An integral operator solution of the YBE (at the plain non-deformed level) was constructed for the first time in [16]. The factorization property of the corresponding R-operator was noticed later in [15], which resulted in a powerful almost purely algebraic techniques of building general R-operators [10,17,19,20].…”