Back to baxterisation
N. Crampe, E. Ragoucy, M. Vanicat
TL;DR
This work extends baxterisation techniques to three new algebras, $\mathcal{A}_{\mathfrak n}(a,b,c)$, $\mathcal{B}_{\mathfrak n}$, and $\mathcal{C}_{\mathfrak n}$, which sit close to the braid algebra and yield $R$-matrices with spectral parameters for the braided Yang–Baxter equation. It establishes how these algebras relate to known cosets, derives scalar and some two-dimensional representations, and presents a main theorem providing explicit forms of the baxterisation function $f(x,y)$ for three cases, enabling braided YBE solutions via $\check R_i(x,y)=(1-f(x,y)\sigma_i)(1-f(y,x)\sigma_i)^{-1}$. The key contributions are the algebra definitions, the scalar and low-dimensional representations, and the explicit baxterisation results for the $\mathcal{A}_3(a,b,c)$, $\mathcal{B}_3$, and $\mathcal{C}_3$ cases, along with a discussion of how these algebras connect to Hecke, BMW, and braid-coset structures. This work expands the toolkit for constructing integrable models by providing new seeds for $R$-matrices and highlights the potential for a broader classification of baxterisable cosets of the braid algebra.
Abstract
In the continuity of our previous paper arXiv:1509.05516, we define three new algebras, $A_{n}(a,b,c)$, $B_{n}$ and $C_{n}$, that are close to the braid algebra. They allow to build solutions to the braided Yang-Baxter equation with spectral parameters. The construction is based on a baxterisation procedure, similar to the one used in the context of Hecke or BMW algebras. The $A_{n}(a,b,c)$ algebra depends on three arbitrary parameters, and when the parameter $a$ is set to zero, we recover the algebra $M_{n}(b,c)$ already introduced elsewhere for purpose of baxterisation. The Hecke algebra (and its baxterisation) can be recovered from a coset of the $A_{n}(0,0,c)$ algebra. The algebra $A_{n}(0,b,-b^2)$ is a coset of the braid algebra. The two other algebras $B_{n}$ and $C_{n}$ do not possess any parameter, and can be also viewed as a coset of the braid algebra.