New solutions to the -invariant Yang–Baxter equations at roots of unity: Cyclic representations

Abstract

We find the all solutions to the sl q (2)-invariant multi-parametric Yang-Baxter equations (YBE) at q = i defined on the cyclic (semi-cyclic, nilpotent) representations of the algebra. We are deriving the solutions in form of the linear combinations over the sl q (2)-invariant objects -projectors. The direct construction of the projector operators at roots of unity gives us an opportunity to consider all the possible cases, including also degenerated one, when the number of the projectors becomes larger, and various type of solutions are arising, and as well as the inhomogeneous case. We are giving a full classification of the YBE solutions for the considered representations. A specific character of the solutions is the existence of the arbitrary functions.

Deinstitutionalization in the Field of Mental Health in Armenia: Contemporary Issues

“…As example, the intertwiner Rmatrices, defined in the theory of the quantum groups, satisfy to the ordinary YBE [14], and in the "check" formulation commute with the generators of the quantum group. There the vector states V constitute the representations of the quantum groups and can differ one from other by the characteristics of the representations -the eigenvalues of the center of the group [14,47]. An interesting example is the case of two-dimensional cyclic states of the sl q (2) group at q 4 [47], where two vector spaces are of the same dimension, but have different characteristics, which bring to various kind of invariant solutions to YBE.…”

“…There the vector states V constitute the representations of the quantum groups and can differ one from other by the characteristics of the representations -the eigenvalues of the center of the group [14,47]. An interesting example is the case of two-dimensional cyclic states of the sl q (2) group at q 4 [47], where two vector spaces are of the same dimension, but have different characteristics, which bring to various kind of invariant solutions to YBE. And one can find how to operate with the inhomogeneous R-matrices in "check" formulation for the usual Yang-Baxter equations, particularly in the works [42,47], where we have presented in the solutions with the quantum group symmetry the permuted projction operators between the representation states.…”

“…In this paper we consider spectral parameter dependent solutions to inhomogeneous GYBE for arbitrary dimensional matrices of two types, which both are the extensions of 4 × 4 eight-vertex trigonometric solution: extension by form (the first kind -X-shaped) and by construction (the second kind). In the work [46] entire description is done of the multiparametric solutions of YBE for the simplest case of 4 × 4 matrices (see also [2,47,48]).…”

Integrability in $[1+d]$ dimensions: combined local equations and commutativity of the transfer matrices

“…The ZTE are the most analyzed and the most productive equations [7,15]. The symmetry properties of the models in 2D give a great advance in the exploring of the integrability (conformal symmetry, quantum group) ( [14]- [26], [48]- [50]). And there are some generalizations for 3D case with the models of the quantum algebra symmetry (e.g.…”

Series of the solutions to Yang–Baxter equations: Hecke type matrices and descendant R-, L-operators

“…For finding the full class of the symmetric YBE solutions it is sufficient to consider the R-matrix in the expansion of the whole basis of invariant operators (projectors), which must be specified for the given set of the representations [12,13,14,28]. In particular, investigating solutions defined on the cyclic and indecomposable representations of the quantum algebra sl q (2) at roots of unity [30,31], we find new solutions and yet a rich variety of the solutions, which are characterised by different structures of decompositions into the projectors and as well by additional (spectral) parameters [34]. As a yet another confirmation of the mentioned observation could be served the existence of a series of solutions to the YBE with symmetry of the quantum super-algebra osp q (1|2) defined on the spin-irreps, which differs from the known solutions [20,22,26], and the discussion done in the Section 2 of this work demonstrates the exact derivation of this series.…”