Askey-Wilson Algebras

AUTHORS:

  • Travis Scrimshaw (2018-08): initial version
class sage.algebras.askey_wilson.AlgebraMorphism(domain, on_generators, position=0, codomain=None, category=None)

Bases: sage.modules.with_basis.morphism.ModuleMorphismByLinearity

An algebra morphism of the Askey-Wilson algebra defined by the images of the generators.

sage.algebras.askey_wilson.AskeyWilsonAlgebra

The (universal) Askey-Wilson algebra.

Let \(R\) be a commutative ring. The universal Askey-Wilson algebra is an associative unital algebra \(\Delta_q\) over \(R[q,q^-1]\) given by the generators \(A, B, C, \alpha, \beta, \gamma\) that satisfy the following relations:

\[\begin{split}\begin{aligned} (q-q^{-1}) \alpha &= (q^2-q^{-2}) A + qBC - q^{-1}CB, \\ (q-q^{-1}) \beta &= (q^2-q^{-2}) B + qCA - q^{-1}AC, \\ (q-q^{-1}) \gamma &= (q^2-q^{-2}) C + qAB - q^{-1}BA. \end{aligned}\end{split}\]

The universal Askey-Wilson contains a Casimir element \(\Omega\), and the elements \(\alpha\), \(\beta\), \(\gamma\), \(\Omega\) generate the center of \(\Delta_q\), which is isomorphic to the polynomial ring \((R[q,q^-1])[\alpha,\beta,\gamma,\Omega]\) (assuming \(q\) is not a root of unity). Furthermore, the relations imply that \(\Delta_q\) has a basis given by monomials \(A^i B^j C^k \alpha^r \beta^s \gamma^t\), where \(i, j, k, r, s, t \in \ZZ_{\geq 0}\).

The universal Askey-Wilson algebra also admits a faithful action of \(PSL_2(\ZZ)\) given by the automorphisms \(\rho\) (permutation_automorphism()):

\[A \mapsto B \mapsto C \mapsto A, \qquad \alpha \mapsto \beta \mapsto \gamma \mapsto \alpha.\]

and \(\sigma\) (reflection_automorphism()):

\[A \mapsto B \mapsto A, C \mapsto C + \frac{AB - BA}{q-q^{-1}}, \qquad \alpha \mapsto \beta \mapsto \alpha, \gamma \mapsto \gamma.\]

Note that \(\rho^3 = \sigma^2 = 1\) and

\[\sigma(C) = C - q AB - (1+q^2) C + q \gamma = C - q AB - q^2 C + q \gamma.\]

The Askey-Wilson \(AW_q(a,b,c)\) algebra is a specialization of the universal Askey-Wilson algebra by \(\alpha = a\), beta = b`, \(\gamma = c\), where \(a,b,c \in R\). \(AW_q(a,b,c)\) was first introduced by [Zhedanov1991] to describe the Askey-Wilson polynomials. The Askey-Wilson algebra has a central extension of \(\Delta_q\).

INPUT:

  • R – a commutative ring
  • q – (optional) the parameter \(q\); must be invertable in R

If q is not specified, then R is taken to be the base ring of a Laurent polynomial ring with variable \(q\). Otherwise the element q must be an element of R.

Note

No check is performed to ensure q is not a root of unity, which may lead to violations of the results in [Terwilliger2011].

EXAMPLES:

We create the universal Askey-Wilson algebra and check the defining relations:

sage: AW = algebras.AskeyWilson(QQ)
sage: AW.inject_variables()
Defining A, B, C, a, b, g
sage: q = AW.q()
sage: (q^2-q^-2)*A + q*B*C - q^-1*C*B == (q-q^-1)*a
True
sage: (q^2-q^-2)*B + q*C*A - q^-1*A*C == (q-q^-1)*b
True
sage: (q^2-q^-2)*C + q*A*B - q^-1*B*A == (q-q^-1)*g
True

Next, we perform some computations:

sage: C * A
(q^-2)*A*C + (q^-3-q)*B - (q^-2-1)*b
sage: B^2 * g^2 * A
q^4*A*B^2*g^2 - (q^-1-q^7)*B*C*g^2 + (1-q^4)*B*g^3
 + (1-2*q^4+q^8)*A*g^2 - (q-q^3-q^5+q^7)*a*g^2
sage: (B^3 - A) * (C^2 + q*A*B)
q^7*A*B^4 + B^3*C^2 - (q^2-q^14)*B^3*C + (q-q^7)*B^3*g - q*A^2*B
 + (3*q^3-4*q^7+q^19)*A*B^2 - A*C^2 - (1-q^6-q^8+q^14)*B^2*a
 - (q^-2-3*q^6+3*q^14-q^22)*B*C
 + (q^-1+q-3*q^3-q^5+2*q^7-q^9+q^13+q^15-q^19)*B*g
 + (2*q^-1-6*q^3+5*q^7-2*q^19+q^23)*A
 - (2-2*q^2-4*q^4+4*q^6+q^8-q^10+q^12-q^14+q^16-q^18-q^20+q^22)*a

We check the elements \(\alpha\), \(\beta\), and \(\gamma\) are in the center:

sage: all(x * gen == gen * x for gen in AW.algebra_generators() for x in [a,b,g])
True

We verify that the Casimir element is in the center:

sage: Omega = AW.casimir_element()
sage: all(x * Omega == Omega * x for x in [A,B,C])
True

sage: x = AW.an_element()
sage: O2 = Omega^2
sage: x * O2 == O2 * x
True

We prove Lemma 2.1 in [Terwilliger2011]:

sage: (q^2-q^-2) * C == (q-q^-1) * g - (q*A*B - q^-1*B*A)
True
sage: (q-q^-1) * (q^2-q^-2) * a == (B^2*A - (q^2+q^-2)*B*A*B + A*B^2
....:                               + (q^2-q^-2)^2*A + (q-q^-1)^2*B*g)
True
sage: (q-q^-1) * (q^2-q^-2) * b == (A^2*B - (q^2+q^-2)*A*B*A + B*A^2
....:                               + (q^2-q^-2)^2*B + (q-q^-1)^2*A*g)
True

We prove Theorem 2.2 in [Terwilliger2011]:

sage: q3 = q^-2 + 1 + q^2
sage: A^3*B - q3*A^2*B*A + q3*A*B*A^2 - B*A^3 == -(q^2-q^-2)^2 * (A*B - B*A)
True
sage: B^3*A - q3*B^2*A*B + q3*B*A*B^2 - A*B^3 == -(q^2-q^-2)^2 * (B*A - A*B)
True
sage: (A^2*B^2 - B^2*A^2 + (q^2+q^-2)*(B*A*B*A-A*B*A*B)
....:  == -(q^1-q^-1)^2 * (A*B - B*A) * g)
True

We construct an Askey-Wilson algebra over \(\GF{5}\) at \(q=2\):

sage: AW = algebras.AskeyWilson(GF(5), q=2)
sage: A,B,C,a,b,g = AW.algebra_generators()
sage: q = AW.q()
sage: Omega = AW.casimir_element()

sage: B * A
4*A*B + 2*g
sage: C * A
4*A*C + 2*b
sage: C * B
4*B*C + 2*a
sage: Omega^2
A^2*B^2*C^2 + A^3*B*C + A*B^3*C + A*B*C^3 + A^4 + 4*A^3*a
 + 2*A^2*B^2 + A^2*B*b + 2*A^2*C^2 + 4*A^2*C*g + 4*A^2*a^2
 + 4*A*B^2*a + 4*A*C^2*a + B^4 + B^3*b + 2*B^2*C^2 + 4*B^2*C*g
 + 4*B^2*b^2 + B*C^2*b + C^4 + 4*C^3*g + 4*C^2*g^2 + 2*a*b*g

sage: (q^2-q^-2)*A + q*B*C - q^-1*C*B == (q-q^-1)*a
True
sage: (q^2-q^-2)*B + q*C*A - q^-1*A*C == (q-q^-1)*b
True
sage: (q^2-q^-2)*C + q*A*B - q^-1*B*A == (q-q^-1)*g
True
sage: all(x * Omega == Omega * x for x in [A,B,C])
True

REFERENCES: