Nil-Coxeter algebra

Definition

The nil-Coxeter algebra for the infinite symmetric group is the algebra generated by u1, u2, u3, ... with the relations

: \begin{align} u_i^2 & = 0, \ u_i u_j & = u_j u_i & & \text{ if } |i-j| 1, \ u_i u_j u_i & = u_j u_i u_j & & \text{ if } |i-j|=1. \end{align}

These are just the relations for the infinite braid group, together with the relations u = 0. Similarly one can define a nil-Coxeter algebra for any Coxeter system, by adding the relations u = 0 to the relations of the corresponding generalized braid group.

References