New solutions to the $s{\ell }_q(2)$-invariant Yang-Baxter equations at roots of unity: cyclic representations

Abstract

We find the all solutions to the $s{l}_q(2)$-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 $s{l}_q(2)$-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.

28 pages; extended version; new references are added

References in corpus (4)

• Characteristics of 2D lattice models from fermionic realization: Ising and $XYZ$ models
• Solutions to the Yang-Baxter equations with $os{p}_q(1|2)$ symmetry: Lax operators
• Fusion Rules of the Lowest Weight Representations of osp_q(1|2) at Roots of Unity: Polynomial Realization and Degeneration at Roots of Unity
• New solutions to the $s{\ell }_q(2)$-invariant Yang-Baxter equations at roots of unity

Cited by in corpus (5)

• On the solutions of the Yang-Baxter equations with general inhomogeneous eight-vertex $R$-matrix: Relations with Zamolodchikov's tetrahedral algebra
• On the solutions to the multi-parametric Yang-Baxter equations
• On Bailey pairs for $\mathcal{N}=2$ supersymmetric gauge theories on $S_b^3/{\mathbb{Z}}_r$
• New series of multi-parametric solutions to GYBE: quantum gates and integrability
• Series of the solutions to Yang-Baxter equations: Hecke type matrices and descendant R-, L-operators