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