Affine nilTemperley Lieb Algebra of type A

sage.algebras.affine_nil_temperley_lieb.AffineNilTemperleyLiebTypeA

Constructs the affine nilTemperley Lieb algebra of type \(A_{n-1}^{(1)}\) as used in [Pos2005].

INPUT:

  • n – a positive integer

The affine nilTemperley Lieb algebra is generated by \(a_i\) for \(i=0,1,\ldots,n-1\) subject to the relations \(a_i a_i = a_i a_{i+1} a_i = a_{i+1} a_i a_{i+1} = 0\) and \(a_i a_j = a_j a_i\) for \(i-j \not \equiv \pm 1\), where the indices are taken modulo \(n\).

EXAMPLES:

sage: A = AffineNilTemperleyLiebTypeA(4)
sage: a = A.algebra_generators(); a
Finite family {0: a0, 1: a1, 2: a2, 3: a3}
sage: a[1]*a[2]*a[0] == a[1]*a[0]*a[2]
True
sage: a[0]*a[3]*a[0]
0
sage: A.an_element()
2*a0 + 1 + 3*a1 + a0*a1*a2*a3