Alternating Central Extension Quantum Onsager Algebra

AUTHORS:

• Travis Scrimshaw (2021-03): Initial version

class sage.algebras.quantum_groups.ace_quantum_onsager.ACEQuantumOnsagerAlgebra(R, q)

Bases: CombinatorialFreeModule

The alternating central extension of the \(q\)-Onsager algebra.

The alternating central extension \(\mathcal{A}_q\) of the \(q\)-Onsager algebra \(O_q\) is a current algebra of \(O_q\) introduced by Baseilhac and Koizumi [BK2005]. A presentation was given by Baseilhac and Shigechi [BS2010], which was then reformulated in terms of currents in [Ter2021] and then used to prove that the generators form a PBW basis.

Note

This is only for the \(q\)-Onsager algebra with parameter \(c = q^{-1} (q - q^{-1})^2\).

EXAMPLES:

sage: A = algebras.AlternatingCentralExtensionQuantumOnsager(QQ)
sage: AG = A.algebra_generators()

We construct the generators \(\mathcal{G}_3\), \(\mathcal{W}_{-5}\), \(\mathcal{W}_2\), and \(\widetilde{\mathcal{G}}_{4}\) and perform some computations:

sage: G3 = AG[0,3]
sage: Wm5 = AG[1,-5]
sage: W2 = AG[1,2]
sage: Gt4 = AG[2,4]
sage: [G3, Wm5, W2, Gt4]
[G[3], W[-5], W[2], Gt[4]]
sage: Gt4 * G3
G[3]*Gt[4] + ((-q^12+3*q^8-3*q^4+1)/q^6)*W[-6]*W[1]
 + ((-q^12+3*q^8-3*q^4+1)/q^6)*W[-5]*W[2]
 + ((q^12-3*q^8+3*q^4-1)/q^6)*W[-4]*W[1]
 + ((-q^12+3*q^8-3*q^4+1)/q^6)*W[-4]*W[3]
 + ((-q^12+3*q^8-3*q^4+1)/q^6)*W[-3]*W[-2]
 + ((q^12-3*q^8+3*q^4-1)/q^6)*W[-3]*W[2]
 + ((q^12-3*q^8+3*q^4-1)/q^6)*W[-2]*W[5]
 + ((-q^12+3*q^8-3*q^4+1)/q^6)*W[-1]*W[4]
 + ((q^12-3*q^8+3*q^4-1)/q^6)*W[-1]*W[6]
 + ((-q^12+3*q^8-3*q^4+1)/q^6)*W[0]*W[5]
 + ((q^12-3*q^8+3*q^4-1)/q^6)*W[0]*W[7]
 + ((q^12-3*q^8+3*q^4-1)/q^6)*W[3]*W[4]
sage: Wm5 * G3
((q^2-1)/q^2)*G[1]*W[-7] + ((-q^2+1)/q^2)*G[1]*W[7]
 + ((q^2-1)/q^2)*G[2]*W[-6] + ((-q^2+1)/q^2)*G[2]*W[6] + G[3]*W[-5]
 + ((-q^2+1)/q^2)*G[6]*W[-2] + ((q^2-1)/q^2)*G[6]*W[2]
 + ((-q^2+1)/q^2)*G[7]*W[-1] + ((q^2-1)/q^2)*G[7]*W[1]
 + ((-q^2+1)/q^2)*G[8]*W[0] + ((-q^8+2*q^4-1)/q^5)*W[-8]
 + ((q^8-2*q^4+1)/q^5)*W[8]
sage: W2 * G3
(q^2-1)*G[1]*W[-2] + (-q^2+1)*G[1]*W[4] + (-q^2+1)*G[3]*W[0]
 + q^2*G[3]*W[2] + (q^2-1)*G[4]*W[1] + ((-q^8+2*q^4-1)/q^3)*W[-3]
 + ((q^8-2*q^4+1)/q^3)*W[5]
sage: W2 * Wm5
(q^4/(q^8+2*q^6-2*q^2-1))*G[1]*Gt[6] + (-q^4/(q^8+2*q^6-2*q^2-1))*G[6]*Gt[1]
 + W[-5]*W[2] + (q/(q^2+1))*G[7] + (-q/(q^2+1))*Gt[7]
sage: Gt4 * Wm5
((q^2-1)/q^2)*W[-8]*Gt[1] + ((q^2-1)/q^2)*W[-7]*Gt[2]
 + ((q^2-1)/q^2)*W[-6]*Gt[3] + W[-5]*Gt[4] + ((-q^2+1)/q^2)*W[-3]*Gt[6]
 + ((-q^2+1)/q^2)*W[-2]*Gt[7] + ((-q^2+1)/q^2)*W[-1]*Gt[8]
 + ((-q^2+1)/q^2)*W[0]*Gt[9] + ((q^2-1)/q^2)*W[1]*Gt[8]
 + ((q^2-1)/q^2)*W[2]*Gt[7] + ((q^2-1)/q^2)*W[3]*Gt[6]
 + ((-q^2+1)/q^2)*W[6]*Gt[3] + ((-q^2+1)/q^2)*W[7]*Gt[2]
 + ((-q^2+1)/q^2)*W[8]*Gt[1] + ((-q^8+2*q^4-1)/q^5)*W[-9]
 + ((q^8-2*q^4+1)/q^5)*W[9]
sage: Gt4 * W2
(q^2-1)*W[-3]*Gt[1] + (-q^2+1)*W[0]*Gt[4] + (q^2-1)*W[1]*Gt[5]
 + q^2*W[2]*Gt[4] + (-q^2+1)*W[5]*Gt[1] + ((-q^8+2*q^4-1)/q^3)*W[-4]
 + ((q^8-2*q^4+1)/q^3)*W[6]

REFERENCES:

• [BK2005]

• [BS2010]

• [Ter2021]

algebra_generators()

Return the algebra generators of self.

EXAMPLES:

sage: A = algebras.AlternatingCentralExtensionQuantumOnsager(QQ)
sage: A.algebra_generators()
Lazy family (generator map(i))_{i in Disjoint union of
 Family (Positive integers, Integer Ring, Positive integers)}

dagger()

The antiautomorphism \(\dagger\).

EXAMPLES:

sage: A = algebras.AlternatingCentralExtensionQuantumOnsager(QQ)
sage: G = A.algebra_generators()
sage: x = A.an_element()^2
sage: A.dagger(A.dagger(x)) == x
True
sage: A.dagger(G[1,-1] * G[1,1]) == A.dagger(G[1,1]) * A.dagger(G[1,-1])
True
sage: A.dagger(G[0,2] * G[1,3]) == A.dagger(G[1,3]) * A.dagger(G[0,2])
True
sage: A.dagger(G[2,2] * G[1,3]) == A.dagger(G[1,3]) * A.dagger(G[2,2])
True

degree_on_basis(m)

Return the degree of the basis element indexed by m.

EXAMPLES:

sage: A = algebras.AlternatingCentralExtensionQuantumOnsager(QQ)
sage: G = A.algebra_generators()
sage: A.degree_on_basis(G[0,1].leading_support())
2
sage: A.degree_on_basis(G[0,2].leading_support())
4
sage: A.degree_on_basis(G[1,-1].leading_support())
3
sage: A.degree_on_basis(G[1,0].leading_support())
1
sage: A.degree_on_basis(G[1,1].leading_support())
1
sage: A.degree_on_basis(G[2,1].leading_support())
2
sage: A.degree_on_basis(G[2,2].leading_support())
4
sage: [x.degree() for x in A.some_elements()]
[1, 5, 3, 1, 5, 2, 4, 2, 4]

gens()

Return the algebra generators of self.

EXAMPLES:

sage: A = algebras.AlternatingCentralExtensionQuantumOnsager(QQ)
sage: A.algebra_generators()
Lazy family (generator map(i))_{i in Disjoint union of
 Family (Positive integers, Integer Ring, Positive integers)}

one_basis()

Return the basis element indexing \(1\).

EXAMPLES:

sage: A = algebras.AlternatingCentralExtensionQuantumOnsager(QQ)
sage: ob = A.one_basis(); ob
1
sage: ob.parent()
Free abelian monoid indexed by Disjoint union of
 Family (Positive integers, Integer Ring, Positive integers)

product_on_basis(lhs, rhs)

Return the product of the two basis elements lhs and rhs.

EXAMPLES:

sage: A = algebras.AlternatingCentralExtensionQuantumOnsager(QQ)
sage: G = A.algebra_generators()
sage: q = A.q()
sage: rho = -(q^2 - q^-2)^2

We verify the PBW ordering:

sage: G[0,1] * G[1,1]  # indirect doctest
G[1]*W[1]
sage: G[1,1] * G[0,1]
q^2*G[1]*W[1] + ((-q^8+2*q^4-1)/q^3)*W[0] + ((q^8-2*q^4+1)/q^3)*W[2]
sage: G[1,-1] * G[1,1]
W[-1]*W[1]
sage: G[1,1] * G[1,-1]
W[-1]*W[1] + (q/(q^2+1))*G[2] + (-q/(q^2+1))*Gt[2]
sage: G[1,1] * G[2,1]
W[1]*Gt[1]
sage: G[2,1] * G[1,1]
q^2*W[1]*Gt[1] + ((-q^8+2*q^4-1)/q^3)*W[0] + ((q^8-2*q^4+1)/q^3)*W[2]
sage: G[0,1] * G[2,1]
G[1]*Gt[1]
sage: G[2,1] * G[0,1]
G[1]*Gt[1] + ((-q^12+3*q^8-3*q^4+1)/q^6)*W[-1]*W[1]
 + ((-q^12+3*q^8-3*q^4+1)/q^6)*W[0]^2
 + ((q^12-3*q^8+3*q^4-1)/q^6)*W[0]*W[2]
 + ((q^12-3*q^8+3*q^4-1)/q^6)*W[1]^2

We verify some of the defining relations (see Equations (3-14) in [Ter2021]), which are used to construct the PBW basis:

sage: G[0,1] * G[0,2] == G[0,2] * G[0,1]
True
sage: G[1,-1] * G[1,-2] == G[1,-2] * G[1,-1]
True
sage: G[1,1] * G[1,2] == G[1,2] * G[1,1]
True
sage: G[2,1] * G[2,2] == G[2,2] * G[2,1]
True
sage: G[1,0] * G[1,2] - G[1,2] * G[1,0] == G[1,-1] * G[1,1] - G[1,1] * G[1,-1]
True
sage: G[1,0] * G[1,2] - G[1,2] * G[1,0] == (G[2,2] - G[0,2]) / (q + ~q)
True
sage: q * G[1,0] * G[0,2] - ~q * G[0,2] * G[1,0] == q * G[2,2] * G[1,0] - ~q * G[1,0] * G[2,2]
True
sage: q * G[1,0] * G[0,2] - ~q * G[0,2] * G[1,0] == rho * G[1,-2] - rho * G[1,2]
True
sage: q * G[0,2] * G[1,1] - ~q * G[1,1] * G[0,2] == q * G[1,1] * G[2,2] - ~q * G[2,2] * G[1,1]
True
sage: q * G[0,2] * G[1,1] - ~q * G[1,1] * G[0,2] == rho * G[1,3] - rho * G[1,-1]
True
sage: G[1,-2] * G[1,2] - G[1,2] * G[1,-2] == G[1,-1] * G[1,3] - G[1,3] * G[1,-1]
True
sage: G[1,-2] * G[0,2] - G[0,2] * G[1,-2] == G[1,-1] * G[0,3] - G[0,3] * G[1,-1]
True
sage: G[1,1] * G[0,2] - G[0,2] * G[1,1] == G[1,2] * G[0,1] - G[0,1] * G[1,2]
True
sage: G[1,-2] * G[2,2] - G[2,2] * G[1,-2] == G[1,-1] * G[2,3] - G[2,3] * G[1,-1]
True
sage: G[1,1] * G[2,2] - G[2,2] * G[1,1] == G[1,2] * G[2,1] - G[2,1] * G[1,2]
True
sage: G[0,1] * G[2,2] - G[2,2] * G[0,1] == G[0,2] * G[2,1] - G[2,1] * G[0,2]
True

q()

Return the parameter \(q\) of self.

EXAMPLES:

sage: A = algebras.AlternatingCentralExtensionQuantumOnsager(QQ)
sage: A.q()
q

quantum_onsager_pbw_generator(i)

Return the image of the PBW generator of the \(q\)-Onsager algebra in self.

INPUT:

• i – a pair (k, m) such that

  • k=0 and m is an integer

  • k=1 and m is a positive integer

EXAMPLES:

sage: A = algebras.AlternatingCentralExtensionQuantumOnsager(QQ)
sage: A.quantum_onsager_pbw_generator((0,0))
W[1]
sage: A.quantum_onsager_pbw_generator((0,1))
(q^3/(q^4-1))*W[1]*Gt[1] - q^2*W[0] + (q^2+1)*W[2]
sage: A.quantum_onsager_pbw_generator((0,2))
(q^6/(q^8-2*q^4+1))*W[1]*Gt[1]^2 + (-q^5/(q^4-1))*W[0]*Gt[1]
 + (q^3/(q^2-1))*W[1]*Gt[2] + (q^3/(q^2-1))*W[2]*Gt[1]
 + (-q^4-q^2)*W[-1] - q^2*W[1] + (q^4+2*q^2+1)*W[3]
sage: A.quantum_onsager_pbw_generator((0,-1))
W[0]
sage: A.quantum_onsager_pbw_generator((0,-2))
(q/(q^4-1))*W[0]*Gt[1] + ((q^2+1)/q^2)*W[-1] - 1/q^2*W[1]
sage: A.quantum_onsager_pbw_generator((0,-3))
(q^2/(q^8-2*q^4+1))*W[0]*Gt[1]^2 + (1/(q^3-q))*W[-1]*Gt[1]
 + (1/(q^3-q))*W[0]*Gt[2] - (1/(q^5-q))*W[1]*Gt[1]
 + ((q^4+2*q^2+1)/q^4)*W[-2] - 1/q^2*W[0] + ((-q^2-1)/q^4)*W[2]
sage: A.quantum_onsager_pbw_generator((1,1))
((-q^2+1)/q^2)*W[0]*W[1] + (1/(q^3+q))*G[1] - (1/(q^3+q))*Gt[1]
sage: A.quantum_onsager_pbw_generator((1,2))
-1/q*W[0]*W[1]*Gt[1] + (1/(q^6+q^4-q^2-1))*G[1]*Gt[1]
 + ((-q^4+1)/q^4)*W[-1]*W[1] + (q^2-1)*W[0]^2
 + ((-q^4+1)/q^2)*W[0]*W[2] + ((q^2-1)/q^4)*W[1]^2
 - (1/(q^6+q^4-q^2-1))*Gt[1]^2 + 1/q^3*G[2] - 1/q^3*Gt[2]

sigma()

The automorphism \(\sigma\).

EXAMPLES:

sage: A = algebras.AlternatingCentralExtensionQuantumOnsager(QQ)
sage: G = A.algebra_generators()
sage: x = A.an_element()^2
sage: A.sigma(A.sigma(x)) == x
True
sage: A.sigma(G[1,-1] * G[1,1]) == A.sigma(G[1,-1]) * A.sigma(G[1,1])
True
sage: A.sigma(G[0,2] * G[1,3]) == A.sigma(G[0,2]) * A.sigma(G[1,3])
True

some_elements()

Return some elements of self.

EXAMPLES:

sage: A = algebras.AlternatingCentralExtensionQuantumOnsager(QQ)
sage: A.some_elements()
[W[0], W[3], W[-1], W[1], W[-2], G[1], G[2], Gt[1], Gt[2]]