Nil-Coxeter Algebra

sage.algebras.nil_coxeter_algebra.NilCoxeterAlgebra

Construct the Nil-Coxeter algebra of given type.

This is the algebra with generators \(u_i\) for every node \(i\) of the corresponding Dynkin diagram. It has the usual braid relations (from the Weyl group) as well as the quadratic relation \(u_i^2 = 0\).

INPUT:

  • W – a Weyl group

OPTIONAL ARGUMENTS:

  • base_ring – a ring (default is the rational numbers)
  • prefix – a label for the generators (default “u”)

EXAMPLES:

sage: U = NilCoxeterAlgebra(WeylGroup(['A',3,1]))
sage: u0, u1, u2, u3 = U.algebra_generators()
sage: u1*u1
0
sage: u2*u1*u2 == u1*u2*u1
True
sage: U.an_element()
u[0,1,2,3] + 2*u[0] + 3*u[1] + 1