Quantum Supergroups and Solutions of the Yang-Baxter Equation

Abstract

A method is developed for systematically constructing trigonometric and rational solutions of the Yang-Baxter equation using the representation theory of quantum supergroups. New quantum R-matrices are obtained by applying the method to the vector representations of quantum osp (1/2) and gl (m/n).

The Bethe ansatz equations for reflecting magnons

“…However, the extension of the proposed method to Z 2 graded algebras is expected to follow the same lines of [37] for the solutions of the graded Yang-Baxter equation based on superalgebras. On the other hand one could have considered the direct resolution of the reflection equation (11) similarly to the analysis performed by Shiroishi and Wadati in [31] for the Hubbard model.…”

The First Fundamental Theorem of Invariant Theory for the Orthosymplectic Supergroup

“…To consider quantum supergroups [4,50,70] from a deformation theoretical point of view [16,17,44,64], we will need the notion of quasi Hopf superalgebras [64,§II], which will be briefly discussed in Appendix B. Let g = g0 ⊕ g1 be a complex Lie superalgebra [22,39] with even subspace g0 and odd subspace g1.…”

“…Quantum supergroups [4,50,70] are a class of quasi-triangular [8] Hopf superalgebras introduced in the early 90s, which have since been studied quite extensively; see e.g., [26,37,52,56,58,60,71] for results on their finite dimensional irreducible representations. Quantum supergroups have been applied to obtain interesting results in several areas, most notably, in the study of Yang-Baxter type integrable models [4,5,67], construction of topological invariants of knots and 3-manifolds [27,57,61,62] and development of quantum supergeometry [63,65].…”

“…Quantum supergroups have been applied to obtain interesting results in several areas, most notably, in the study of Yang-Baxter type integrable models [4,5,67], construction of topological invariants of knots and 3-manifolds [27,57,61,62] and development of quantum supergeometry [63,65]. It is the quasi triangular Hopf superalgebraic structure [19,25,50] that renders quantum supergroups so useful in so many areas.…”

“…It was shown in [64] (also see [16,17]) that the universal enveloping superalgebra U(g; T ) over the power series ring T = C [[t]] can be endowed with the structure of a braided quasi Hopf superalgebra, where the universal R-matrix was constructed from the quadratic Casimir operator and the associator was obtained by using solutions of a KZ equation following the strategy of Drinfeld [9]. The Drinfeld-Jimbo quantum supergroup U q (g; T ) [4,50,70] over T is the Etingof-Kazhdan quantisation of U(g; T ) [16,17]. It follows from a general property of the Etingof-Kazhdan quantisation that U q (g; T ) and U(g; T ) are equivalent (cf.…”