In mathematics, the nil-Coxeter algebra, introduced by Fomin & Stanley (1994), is an algebra similar to the group algebra of a Coxeter group except that the generators are nilpotent.
The nil-Coxeter algebra for the infinite symmetric group is the algebra generated by u1, u2, u3, ... with the relations These are just the relations for the infinite braid group, together with the relations u2i = 0.
Similarly one can define a nil-Coxeter algebra for any Coxeter system, by adding the relations u2i = 0 to the relations of the corresponding generalized braid group.