In mathematics, the nil-Coxeter algebra, introduced by , is an algebra similar to the group algebra

Group algebra

In mathematics, the group algebra is any of various constructions to assign to a locally compact group an operator algebra , such that representations of the algebra are related to representations of the group...

 of a Coxeter group

Coxeter group

In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...

 except that the generators are nilpotent

Nilpotent

In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0....

.

Definition

The nil-Coxeter algebra for the infinite symmetric group

Symmetric group

In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...

 is the algebra generated by u₁, u₂, u₃, ... with the relations

• u = 0
• u_iu_j = u_ju_i if |i − j| > 1
• u_iu_ju_i = u_ju_iu_j if |i − j| = 1

These are just the relations for the infinite braid group

Braid group

In mathematics, the braid group on n strands, denoted by Bn, is a group which has an intuitive geometrical representation, and in a sense generalizes the symmetric group Sn. Here, n is a natural number; if n > 1, then Bn is an infinite 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.