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

TL;DR

This paper classifies all solutions to the $sl_q(2)$-invariant Yang-Baxter equations at roots of unity, focusing on cyclic and related representations, revealing solutions with arbitrary functions and including degenerate cases.

Contribution

It provides a comprehensive classification of solutions to the Yang-Baxter equations at roots of unity using projector operators, including degenerate and inhomogeneous cases.

Findings

Complete set of solutions including degenerate cases

Solutions expressed as linear combinations of projectors

Existence of arbitrary functions in solutions

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.