Lax Operator for the Quantised Orthosymplectic Superalgebra U q [osp(m|n)]

Abstract

Representations of quantum superalgebras provide a natural framework in which to model supersymmetric quantum systems. Each quantum superalgebra, belonging to the class of quasitriangular Hopf superalgebras, contains a universal R-matrix which automatically satisfies the Yang-Baxter equation. Applying the vector representation π, which acts on the vector module V , to the left-hand side of a universal R-matrix gives a Lax operator. In this Communication a Lax operator is constructed for the quantised orthosymplectic superalgebras Uq[osp(m|n)] for all m > 2, n ≥ 0 where n is even. This can then be used to find a solution to the Yang-Baxter equation acting on V ⊗ V ⊗ W , where W is an arbitrary Uq[osp(m|n)] module. The case W = V is studied as an example.

Solutions to the Yang–Baxter equations with symmetry: Lax operators

“…For completeness sake we want to mention the work [19], where the authors connect the quest of the Lax operator of osp q (1|2n) with the isomorphism existing between the representations of osp q (1|2n) and so q (2n + 1) [10]. Our investigation in principle differs from it, as we consider and manipulate directly the osp q (1|2) symmetric matrices.…”

“…In this paper we are constructing the simple form of the Lax operator and are analyzing the all osp q (1|2) invariant solutions to the spectral parameter dependent YBE. For completeness sake we want to mention the work [19], where the authors connect the quest of the Lax operator of osp q (1|2n) with the isomorphism existing between the representations of osp q (1|2n) and so q (2n + 1) [10]. Our investigation in principle differs from it, as we consider and manipulate directly the osp q (1|2) symmetric matrices.…”

Eigenvalues of Casimir invariants for Uq[osp(m∣n)]

“…In this paper we construct the Casimir invariants (central elements) of quantised orthosymplectic superalgebras. Our method of construction follows from the general results of [10] and the explicit form of the Lax operator obtained in [11]. A fundamental problem is to determine the eigenvalues of the Casimir invariants when acting on an arbitrary finite-dimensional irreducible module.…”

“…The quantum superalgebra U q [osp(m|n)] is a q-deformation of the classical orthosymplectic superalgebra. A brief explanation of U q [osp(m|n)] is given below, with a more thorough introduction to osp(m|n) and the q-deformation to be found in [11].…”

“…To define the opposite Lax operator R T = (π ⊗ id)R we require the graded conjugation action †, which is defined on the simple generators by (see [11]):…”