R-Matrix and Baxter Q-Operators for the Noncompact SL(N,C) Invariant Spin Chain

Abstract

Abstract. The problem of constructing the SL(N, C) invariant solutions to the YangBaxter equation is considered. The solutions (R-operators) for arbitrarily principal series representations of SL(N, C) are obtained in an explicit form. We construct the commutative family of the operators Q k (u) which can be identified with the Baxter operators for the noncompact SL(N, C) spin magnet.

Yang–Baxter -operators and parameter permutations

“…There are 2n parameters in the RLL-relation and the permutation group is now S 2n . There are 2n − 1 elementary transpositions in the group S 2n The R-matrix again admits the decomposition in factors of operators representing the elementary transpositions [17]. The operators, representing s 1 , · · · , s n−1 and s n+1 , · · · , s 2n−1 are the well known intertwining operators and are connected to the Weyl group of the algebra sℓ(n).…”

“…but now the Lax matrix depends on the n parameters u 1 , u 2 · · · u n [17]. There are n − 1 parameters which label the representation of the algebra sℓ(n) (analog of spin ℓ) and spectral parameter u.…”

“…The operators, representing s 1 , · · · , s n−1 and s n+1 , · · · , s 2n−1 are the well known intertwining operators and are connected to the Weyl group of the algebra sℓ(n). The operator representing s n plays a special role and can be constructed explicitly [17].…”

Operator approach to analytical evaluation of Feynman diagrams

“…The proposed operator relations (3.6), (4.17) not only clarify the structure of the integrable Lipatov model (see subsection 4.2), but also help to investigate certain generalizations of this model (see [22]). …”

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

“…In the paper [46], an R-matrix acting in the tensor product of two continuous series representations of the group SL(n, C) was constructed. In the case of the group SL(2, C), this was done in [21].…”