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.
The nil-Coxeter algebra for the infinite symmetric group is the algebra generated by u<sub>1</sub>, u<sub>2</sub>, u<sub>3</sub>, ... with the relations
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.