PBW-deformations of Graded Algebras with Braiding Relations

Abstract

The aim of this paper is to describe all PBW-deformations of the connected graded \({\mathbb {K}}\)-algebra \(\mathcal {A}\) generated by \(x_i, 1\le i\le n,\) with the braiding relations:

$$\begin{aligned} \left\{ \begin{array}{ll} x_i^2=0, \ 1\le i\le n, \\ x_ix_j=x_jx_i, \ {|j-i|} >1, \\ x_ix_{i+1}x_i=x_{i+1}x_ix_{i+1}, \ 1\le i\le n-1. \end{array}\right. \end{aligned}$$

Firstly, the complexity \(\mathcal {C}({\mathcal {A}})\) of the algebra \({\mathcal {A}}\) is computed. Then all PBW-deformations of \(\mathcal {A}\) when \(n\ge 2\) are given explicitly with the help of the general PBW-deformation theory introduced by Cassidy and Shelton. Finally, it is shown that each non-trivial PBW-deformation of \(\mathcal {A}\) is isomorphic to a Iwahori-Hecke algebra \(H_q(n+1)\) (of type A) with n generators and an appropriate parameter q. Here, trivial PBW-deformations of \({\mathcal {A}}\) mean that those PBW-deformations that are isomorphic to \({\mathcal {A}}.\)