Nil-Coxeter algebra
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.
Definition
The nil-Coxeter algebra for the infinite symmetric group is the algebra generated by u1, u2, u3, ... with the relations
- u2
i = 0 - uiuj = ujui if |i − j| > 1
- uiujui = ujuiuj if |i − j| = 1
These are just the relations for the infinite braid group, together with the relations u2
i = 0. Similarly one can define a nil-Coxeter algebra for any Coxeter system, by adding the relations u2
i = 0 to the relations of the corresponding generalized braid group.