Abstract
In the continuity of our previous paper (Crampe et al. in Commun Math Phys 349:271, 2017, arXiv:1509.05516 ), we define three new algebras, ${\mathcal{A}_{\mathfrak{n}}(a,b,c)}$ , ${\mathcal{B}_{\mathfrak{n}}}$ and ${\mathcal{C}_{\mathfrak{n}}}$ , that are close to the braid algebra. They allow to build solutions to the 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 ${\mathcal{A}_{\mathfrak{n}}(a,b,c)}$ algebra depends on three arbitrary parameters, and when the parameter a is set to zero, we recover the algebra ${\mathcal{M}_{\mathfrak{n}}(b,c)}$ already introduced elsewhere for purpose of baxterisation. The Hecke algebra (and its baxterisation) can be recovered from a coset of the ${\mathcal{A}_{\mathfrak{n}}(0,0,c)}$ algebra. The algebra ${\mathcal{A}_{\mathfrak{n}}(0,b,-b^2)}$ is a coset of the braid algebra. The two other algebras ${\mathcal{B}_{\mathfrak{n}}}$ and ${\mathcal{C}_{\mathfrak{n}}}$ do not possess any parameter, and can be also viewed as a coset of the braid algebra.