R-matrices and spectrum of vertex models based on superalgebras

Abstract

In this paper we investigate trigonometric vertex models associated with solutions of the Yang-Baxter equation which are invariant relative to q-deformed superalgebras sl(r|2m) (2) , osp(r|2m) (1) and osp(r = 2n|2m) (2) . The associated R-matrices are presented in terms of the standard Weyl basis making possible the formulation of the quantum inverse scattering method for these lattice models. This allowed us to derive the eigenvectors and the eigenvalues of the corresponding transfer matrices as well as explicit expressions for the Bethe ansatz equations.

New R-matrices from representations of braid-monoid algebras based on superalgebras

“…and U q [osp(r = 2n|2m) (2) ] R-matrices previously obtained by us [14]. These generalized representations are shown to satisfy the Birman-Wenzl-Murakami algebra for a variety of grading choices.…”

“…In spite of these difficulties, some progresses have recently been made towards to the presentation of explicit expressions for the R-matrices based on general classes of superalgebras [14,15]. We have for instance exhibited [14] the R-matrices associated to the U q [sl(r|2m) (2) ], U q [osp(r|2m) (1) ] and U q [osp(r = 2n|2m) (2) ] quantum superalgebras in terms of the Weyl matrices.…”

“…We have for instance exhibited [14] the R-matrices associated to the U q [sl(r|2m) (2) ], U q [osp(r|2m) (1) ] and U q [osp(r = 2n|2m) (2) ] quantum superalgebras in terms of the Weyl matrices. This step makes possible the statistical mechanics interpretation of these systems and has offered suitable expressions to perform the corresponding transfer matrices diagonalization.…”

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

“…H.Saleur in 1990 published solution to the spectral parameter dependent YBE for the universal homogeneous R matrix with the quantum deformation of the osp(1|2) symmetry [8]. Since that time different authors turned repeatedly to the study of the osp q (1|2) superalgebra and integrable models with osp q (1|2) symmetry (see for example [9,10,11,12,13,14,15,16,17,18]), but some questions are unclear so far. In particular how many different fundamental R matrices with osp q (1|2) exist?…”

On the multiparametric ${\mathcal{U}}_{q}[{D}_{n+1}^{(2)}]$ vertex model

“…This quantum group approach permits us to reduce the problem (1) to a linear one, in order to associate a fundamental trigonometric Rmatrix to each Lie algebra [6,7] or Lie superalgebra [8,9,10].…”