Back to baxterisation

TL;DR

This paper introduces three new algebras related to the braid algebra, enabling solutions to the braided Yang-Baxter equation with spectral parameters through a baxterisation process, extending previous algebraic frameworks.

Contribution

It defines three new algebras, $A_{n}(a,b,c)$, $B_{n}$, and $C_{n}$, that generalize the braid algebra and facilitate baxterisation for spectral parameter solutions.

Findings

Defined three new algebras related to braid algebra.

Established baxterisation procedures for these algebras.

Connected new algebras to known structures like Hecke and BMW algebras.

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.