The Universal Askey-Wilson Algebra

In 1992 A. Zhedanov introduced the Askey-Wilson algebra AW=AW(3) and used it to describe the Askey-Wilson polynomials. In this paper we introduce a central extension $\Delta$ of AW, obtained from AW by reinterpreting certain parameters as central elements in the algebra. We call $\Delta$ the {\it universal Askey-Wilson algebra}. We give a faithful action of the modular group ${\rm {PSL}}_2({\mathbb Z})$ on $\Delta$ as a group of automorphisms. We give a linear basis for $\Delta$. We describe the center of $\Delta$ and the 2-sided ideal $\Delta[\Delta,\Delta]\Delta$. We discuss how $\Delta$ is related to the $q$-Onsager algebra.


Motivation: Leonard pairs
We recall the notion of a Leonard pair. To do this, we first recall what it means for a matrix to be tridiagonal.
The following matrices are tridiagonal. Tridiagonal means each nonzero entry lies on either the diagonal, the subdiagonal, or the superdiagonal.
The tridiagonal matrix on the left is irreducible. This means each entry on the subdiagonal is nonzero and each entry on the superdiagonal is nonzero.

The Definition of a Leonard Pair
We now define a Leonard pair. From now on F will denote a field.
Definition (Terwilliger 1999) Let V denote a vector space over F with finite positive dimension. By a Leonard pair on V , we mean a pair of linear transformations A : V → V and B : V → V such that: 1 There exists a basis for V with respect to which the matrix representing A is diagonal and the matrix representing B is irreducible tridiagonal.
2 There exists a basis for V with respect to which the matrix representing B is diagonal and the matrix representing A is irreducible tridiagonal.

Leonard pairs in summary
In summary, for a Leonard pair A, B

Leonard pair example
Here is an example of a Leonard pair.
Leonard pair example, cont. Define Leonard pair example, cont.
The Z 3 -symmetric Askey-Wilson relations Theorem (Hau-wen Huang 2011) Referring to the above Leonard pair A, B of QRacah type, there exists an element C such that The above equations are called the Z 3 -symmetric Askey-Wilson relations.

Leonard triples
In the previous example the Z 3 -symmetry involving A, B, C suggests that we should consider Leonard triples along with Leonard pairs.
The notion of a Leonard triple is due to Brian Curtin and defined as follows.

Definition (Brian Curtin 2007)
By a Leonard triple on V we mean an ordered triple of linear transformations (A, B, C ) in End(V ) such that for each φ ∈ {A, B, C } there exists a basis for V with respect to which the matrix representing φ is diagonal and the matrices representing the other two linear transformations are irreducible tridiagonal.

Leonard pairs and Leonard triples
For a moment let us return to our Leonard pair A, B of QRacah type.
Consider the element C from the Z 3 -symmetric Askey-Wilson relations.
Huang has found necessary and sufficient conditions on C for the triple A, B, C to be a Leonard triple. This is explained in the next two theorems.

The universal Askey-Wilson algebra
We now define a central extension of AW, called the universal Askey-Wilson algebra and denoted ∆.
The algebra ∆ involves just one parameter q.
The algebra ∆ is defined as follows.
For the rest of the talk, q denotes a nonzero scalar in F such that q 4 = 1.
The universal Askey-Wilson algebra

Definition (Ter 2011)
Define an F-algebra ∆ = ∆ q by generators and relations in the following way. The generators are A, B, C . The relations assert that each of We call ∆ the universal Askey-Wilson algebra.

The universal Askey-Wilson algebra
By constuction, each Askey-Wilson algebra AW is a homomorphic image of ∆.
By construction, each Leonard pair or triple of QRacah type can be viewed as a ∆-module.

∆ and Q-polynomial DRGs
We now briefly relate ∆ to Q-polynomial distance-regular graphs.
Let Γ denote a distance-regular graph with diameter D ≥ 3 and distance matrices Assume that the Q-polynomial structure has QRacah type; this means (in the notation of Bannai/Ito) type I with each of s, s * nonzero.
Assume that each irreducible T -module is thin. Here T = T (x) is the subconstituent algebra of Γ with respect to x, generated by A 1 and A * 1 .

Theorem (Arjana Zitnik, Ter, in preparation)
With the above assumptions and notation, there exists a surjective algebra homomorphism ∆ → T that sends the generator A to a linear combination of I , A 1 and the generator B to a linear combination of I , A * 1 .

Three central elements of ∆
We now describe ∆ from a ring theoretic point of view.

Definition
Define elements α, β, γ of ∆ such that Note that each of α, β, γ is central in ∆.

Theorem (Ter 2011)
The following is a basis for the F-vector space ∆: We proved this using the Bergman Diamond Lemma.
An action of PSL 2 (Z) on ∆ Recall that the modular group PSL 2 (Z) has a presentation by generators p, s and relations p 3 = 1, s 2 = 1.
Our next goal is to show that PSL 2 (Z) acts on ∆ as a group of automorphisms.
Strategy: identify two automorphisms of ∆ that have orders 3 and 2.
By construction ∆ has an automorphism that sends This automorphism has order 3.
To find an automorphism of ∆ that has order 2, we use another presentation for ∆.

Alternate presentation for ∆
Theorem (Ter 2011) The algebra ∆ has a presentation by generators A, B, γ and relations Here [n] q = (q n − q −n )/(q − q −1 ).
The first two relations above are the tridiagonal relations. following way: This action is faithful.
The Casimir element Ω of ∆ Shortly we will describe the center Z (∆).
To do this we introduce a certain element Ω ∈ ∆ called the Casimir element.

Definition
Define We call Ω the Casimir element of ∆.
The algebra ∆ and U q (sl 2 ) Our next goal is to explain how ∆ is related to the quantum group U q (sl 2 ).

Definition
The F-algebra U = U q (sl 2 ) is defined by generators e, f , k ±1 and relations We call e, f , k ±1 the Chevalley generators for U.
Irreducible modules for U q (sl 2 ) We review the finite-dimensional irreducible modules for U q (sl 2 ).

Lemma
For all integers d ≥ 0 and ε ∈ {1, −1} there exists a U-module V d,ε with the following property: V d,ε has a basis {v i } d i=0 such that The U-module V d,ε is irreducible provided that q is not a root of unity.
The Casimir element of U q (sl 2 ), cont.
The following result is well known.

Lemma
The Casimir element Λ is in the center Z (U). Moreover on the U-module V d,ε The equitable presentation of U q (sl 2 ) When we defined U we used the Chevalley presentation. There is another presentation for U of interest, said to be equitable.

Lemma (Tatsuro Ito, Chih-wen Weng, Ter 2000)
The algebra U has a presentation by generators x, y ±1 , z and relations We call x, y ±1 , z the equitable generators for U.
The U-module V d,ε from the equitable point of view In the equitable presentation the U-module V d,ε looks as follows.
V d,ε has three bases such that: x y z basis 1 diagonal lower bidiagonal upper bidiagonal basis 2 upper bidiagonal diagonal lower bidiagonal basis 3 lower bidiagonal upper bidiagonal diagonal The Casimir element Λ of U q (sl 2 ) In the equitable presentation of U the Casimir element looks as follows.
∆ and U q (sl 2 ) We are now ready to describe how ∆ is related to U q (sl 2 ).

Lemma (Ter 2011)
Let a, b, c denote nonzero scalars in F. Then there exists an F-algebra homomorpism ∆ → U q (sl 2 ) that sends The above homomorphism is not injective. To shrink the kernel we do the following.
Our next goal is to describe how ∆ is related to the double affine Hecke algebra (DAHA) of type (C ∨ 1 , C 1 ). This is the most general DAHA of rank 1.
We will work with the "universal" version of DAHA.
For notational convenience define a four element set The universal DAHA of type (C ∨ 1 , C 1 )

Definition
LetĤ q denote the F-algebra defined by generators {t ±1 i } i∈I and relations We callĤ q the universal DAHA of type (C ∨ 1 , C 1 ).
For notational convenience define The elements X , Y ofĤ q We will describe how ∆ is related toĤ q .
To set the stage we first mention a few basic features ofĤ q . Define Note that X , Y are invertible.
The Artin braid group B 3 We will be discussing the Artin braid group B 3 .
There exists a group homomorphism B 3 → PSL 2 (Z) that sends ρ → p and σ → s. Via this homomorphism we pull back the PSL 2 (Z) action on ∆, to get a B 3 action on ∆ as a group of automorphisms.
Next we explain how B 3 acts onĤ q as a group of automorphisms.

Theorem (Ter 2012)
Under the homomorphism ψ the image of the Casimir element Ω is ∆ andĤ q

Theorem (Ter 2012)
For all g ∈ B 3 the following diagram commutes:

Notation
For notational convenience, from now on identify ∆ with its image under the injection ψ : ∆ →Ĥ q .

From this point of view
A presentation for the spherical subalgebra by generators and relations We now give a presentation of the spherical subalgebra {h ∈Ĥ q |t 0 h = ht 0 } by generators and relations.
This will be our last result of the talk.
A presentation for the spherical subalgebra

Theorem (Ter 2012)
The spherical subalgebra {h ∈Ĥ q |t 0 h = ht 0 } is presented by generators and relations in the following way. The generators are A, B, C , t ±1 0 , {T i } 3 i=1 . The relations assert that each of t ±1 0 ,

Summary
In this talk we introduced the universal Askey-Wilson algebra ∆.
We showed how each Leonard pair and Leonard triple of QRacah type yields a ∆-module.
We discussed how ∆ is related to Q-polynomial distance-regular graphs of QRacah type.
We gave several bases for ∆, we described its center, and we showed how PSL 2 (Z) acts on ∆ as a group of automorphisms.
We described how ∆ is related to U q (sl 2 ).
Finally we described how ∆ is related to the universal DAHA of type (C ∨ 1 , C 1 ). Thank you for your attention!