Nil-Coxeter algebra explained

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 ^{2} _{i} & = 0 , \\ u _{i} u _{j} & = u _{j} u _{i} & & if |i - j| > 1 , \\ u _{i} u _{j} u _{i} & = u _{j} u _{i} u _{j} & & 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.