Nil-Coxeter algebra

In mathematics, the nil-Coxeter algebra, introduced by, is an algebra similar to the group algebra of a Coxeter group except that the generators are nilpotent.

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$2 i$ = 0. Similarly one can define a nil-Coxeter algebra for any Coxeter system, by adding the relations u$2 i$ = 0 to the relations of the corresponding generalized braid group.