The universal Askey-Wilson algebra
Paul Terwilliger
University of Wisconsin-Madison
Paul Terwilliger
The universal Askey-Wilson algebra
Overview
This talk concerns an algebra ∆ called the Universal
Askey-Wilson algebra.
As we will see, ∆ is related to:
• Leonard pairs and Leonard triples of QRacah type
• Q-polynomial distance-regular graphs of QRacah type
• The modular group PSL2 (Z)
• The equitable presentation of the quantum group Uq (sl2 )
• The double affine Hecke algebra of type (C1∨ , C1 )
Paul Terwilliger
The universal Askey-Wilson 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.

2
 1

 0
0
3
4
5
0
0
2
3
3

0
0 
,
3 
0

2
 0

 0
0
3
4
2
0
0
2
1
1

0
0 
.
0 
5
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.
Paul Terwilliger
The universal Askey-Wilson algebra
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.
Paul Terwilliger
The universal Askey-Wilson algebra
Leonard pairs in summary
In summary, for a Leonard pair A, B
basis 1
basis 2
A
diagonal
irred. tridiagonal
Paul Terwilliger
B
irred. tridiagonal
diagonal
The universal Askey-Wilson algebra
Leonard pairs and orthogonal polynomials
The term Leonard pair is motivated by a 1982 theorem of Doug
Leonard concerning the QRacah polynomials and some related
polynomials in the Askey scheme.
For a detailed version of Leonard’s theorem see the book
E. Bannai and T. Ito. Algebraic Combinatorics I: Association
Schemes. Benjamin Cummings. London 1984.
Paul Terwilliger
The universal Askey-Wilson algebra
Leonard pair example
Here is an example of a Leonard pair.
Fix an integer d ≥ 3.
Pick nonzero scalars a, b, c, q in F such that
(i) q 2i 6= 1 for 1 ≤ i ≤ d;
(ii) Neither of a2 , b 2 is among q 2d−2 , q 2d−4 , . . . , q 2−2d ;
(iii) None of abc, a−1 bc, ab −1 c, abc −1 is among
q d−1 , q d−3 , . . . , q 1−d .
Paul Terwilliger
The universal Askey-Wilson algebra
Leonard pair example, cont.
Define
θi
= aq 2i−d + a−1 q d−2i ,
θi∗ = bq 2i−d + b −1 q d−2i
for 0 ≤ i ≤ d and
ϕi
= a−1 b −1 q d+1 (q i − q −i )(q i−d−1 − q d−i+1 )
(q −i − abcq i−d−1 )(q −i − abc −1 q i−d−1 )
for 1 ≤ i ≤ d.
Paul Terwilliger
The universal Askey-Wilson algebra
Leonard pair example, cont.
Define

θ0
 1 θ1


1 θ2
A = 

· ·


·
0
 ∗
θ0 ϕ1

θ1∗ ϕ2


θ2∗ ·
B = 

·


0
Paul Terwilliger
0
·
1 θd
0














·

· ϕd 
θd∗
The universal Askey-Wilson algebra
Leonard pair example cont.
Then the pair A, B is a Leonard pair on the vector space
V = Fd+1 .
A Leonard pair of this form is said to have QRacah type.
This is the most general type of Leonard pair.
Paul Terwilliger
The universal Askey-Wilson algebra
Leonard pairs of QRacah type
Hau-wen Huang (former student of Chih-wen Weng in the
Department of Applied Math, National Chiao Tung University,
Taiwan) has proven the following beautiful theorem about the
Leonard pairs of QRacah type.
Paul Terwilliger
The universal Askey-Wilson algebra
The Z3 -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
(a + a−1 )(q d+1 + q −d−1 ) + (b + b −1 )(c + c −1 )
qBC − q −1 CB
=
I,
2
−2
q −q
q + q −1
(b + b −1 )(q d+1 + q −d−1 ) + (c + c −1 )(a + a−1 )
qCA − q −1 AC
=
I,
B+
2
−2
q −q
q + q −1
qAB − q −1 BA
(c + c −1 )(q d+1 + q −d−1 ) + (a + a−1 )(b + b −1 )
=
I.
C+
2
−2
q −q
q + q −1
A+
The above equations are called the Z3 -symmetric Askey-Wilson
relations.
Paul Terwilliger
The universal Askey-Wilson algebra
Leonard triples
In the previous example the Z3 -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.
Paul Terwilliger
The universal Askey-Wilson algebra
Leonard triples in summary
In summary, for a Leonard triple A, B, C
basis 1
basis 2
basis 3
A
diagonal
irred. tridiagonal
irred. tridiagonal
B
irred. tridiagonal
diagonal
irred. tridiagonal
Paul Terwilliger
C
irred. tridiagonal
irred. tridiagonal
diagonal
The universal Askey-Wilson algebra
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 Z3 -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.
Paul Terwilliger
The universal Askey-Wilson algebra
Leonard pairs and Leonard triples, cont.
Theorem (Hau-wen Huang 2011)
The roots of the characteristic polynomial of C are {θiε }di=0 , where
θiε = cq 2i−d + c −1 q d−2i
Paul Terwilliger
(0 ≤ i ≤ d)
The universal Askey-Wilson algebra
Leonard pairs and Leonard triples, cont.
Theorem (Hau-wen Huang 2011)
The following (i)–(iii) are equivalent.
(i) The triple A, B, C is a Leonard triple;
(ii) {θiε }di=0 are mutually distinct;
(iii) c 2 is not among q 2d−2 , q 2d−4 , . . . , q 2−2d .
The above Leonard triple is said to have QRacah type.
Paul Terwilliger
The universal Askey-Wilson algebra
The Askey-Wilson algebra
In 1992 Alexei Zhedanov introduced the Askey-Wilson algebra
AW=AW(3) and used it to describe the Askey-Wilson polynomials.
Essentially, AW is the algebra defined by three generators A, B, C
subject to the Z3 -symmetric Askey-Wilson relations (The original
definition was somewhat different).
The algebra AW is defined using four parameters a, b, c, q.
Paul Terwilliger
The universal Askey-Wilson algebra
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 6= 1.
Paul Terwilliger
The universal Askey-Wilson algebra
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
A+
qBC − q −1 CB
,
q 2 − q −2
B+
qCA − q −1 AC
,
q 2 − q −2
C+
qAB − q −1 BA
q 2 − q −2
is central in ∆. We call ∆ the universal Askey-Wilson algebra.
Paul Terwilliger
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.
Paul Terwilliger
The universal Askey-Wilson algebra
∆ 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 {Ai }D
i=0 .
Assume Γ has a Q-polynomial ordering {Ei }D
i=0 of its primitive
idempotents.
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.
Paul Terwilliger
The universal Askey-Wilson algebra
∆ and Q-polynomial DRGs, cont.
Fix a vertex x of Γ. For 0 ≤ i ≤ D let A∗i = A∗i (x) denote the dual
distance matrix of Γ that corresponds to Ei and x.
Assume that each irreducible T -module is thin. Here T = T (x) is
the subconstituent algebra of Γ with respect to x, generated by A1
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 , A1 and the generator B to a linear
combination of I , A∗1 .
Paul Terwilliger
The universal Askey-Wilson algebra
Three central elements of ∆
We now describe ∆ from a ring theoretic point of view.
Definition
Define elements α, β, γ of ∆ such that
qBC − q −1 CB
q 2 − q −2
qCA − q −1 AC
B+
q 2 − q −2
qAB − q −1 BA
C+
q 2 − q −2
A+
=
=
=
α
,
q + q −1
β
,
q + q −1
γ
.
q + q −1
Note that each of α, β, γ is central in ∆.
Paul Terwilliger
The universal Askey-Wilson algebra
A basis for ∆
Theorem (Ter 2011)
The following is a basis for the F-vector space ∆:
Ai B j C k αr β s γ t
i, j, k, r , s, t ∈ N.
We proved this using the Bergman Diamond Lemma.
Paul Terwilliger
The universal Askey-Wilson algebra
An action of PSL2 (Z) on ∆
Recall that the modular group PSL2 (Z) has a presentation by
generators p, s and relations p 3 = 1, s 2 = 1.
Our next goal is to show that PSL2 (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
A 7→ B 7→ C 7→ A.
This automorphism has order 3.
To find an automorphism of ∆ that has order 2, we use another
presentation for ∆.
Paul Terwilliger
The universal Askey-Wilson algebra
Alternate presentation for ∆
Theorem (Ter 2011)
The algebra ∆ has a presentation by generators A, B, γ and
relations
A3 B − [3]q A2 BA + [3]q ABA2 − BA3 = −(q 2 − q −2 )2 (AB − BA),
B 3 A − [3]q B 2 AB + [3]q BAB 2 − AB 3 = −(q 2 − q −2 )2 (BA − AB),
A2 B 2 − B 2 A2 + (q 2 + q −2 )(BABA − ABAB)
= −(q − q −1 )2 (AB − BA)γ,
γA = Aγ,
γB = Bγ.
Here [n]q = (q n − q −n )/(q − q −1 ).
The first two relations above are the tridiagonal relations.
Paul Terwilliger
The universal Askey-Wilson algebra
An automorphism of ∆ that has order 2
By the alternate presentation ∆ has an automorphism that swaps
A, B and fixes γ.
This automorphism has order 2.
Paul Terwilliger
The universal Askey-Wilson algebra
An action of PSL2 (Z) on ∆
Theorem (Ter 2011)
The group PSL2 (Z) acts on ∆ as a group of automorphisms in the
following way:
u
p(u)
s(u)
A
B
B
B
C
A
C
A
C+
AB−BA
q−q −1
α
β
β
β
γ
α
γ
α
γ
This action is faithful.
Paul Terwilliger
The universal Askey-Wilson algebra
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
Ω = q −1 ACB + q −2 A2 + q −2 B 2 + q 2 C 2 − q −1 Aα − q −1 Bβ − qC γ.
We call Ω the Casimir element of ∆.
Paul Terwilliger
The universal Askey-Wilson algebra
The Casimir element Ω is central
Theorem (Ter 2011)
The Casimir element Ω is contained in Z (∆).
Moreover Ω is fixed by everything in PSL2 (Z).
Paul Terwilliger
The universal Askey-Wilson algebra
A basis for ∆ that involves Ω
We are going to show that Z (∆) is generated by Ω, α, β, γ
provided that q is not a root of unity.
To this end we display a basis for ∆ that involves Ω.
Lemma (Ter 2011)
The following is a basis for the F-vector space ∆:
Ai B j C k Ω` αr β s γ t
i, j, k, `, r , s, t ∈ N
Paul Terwilliger
ijk = 0.
The universal Askey-Wilson algebra
Algebraic independence
Corollary (Ter 2011)
The elements Ω, α, β, γ are algebraically independent over F.
Paul Terwilliger
The universal Askey-Wilson algebra
The center Z (∆)
We now describe the center Z (∆).
Theorem (Ter 2011)
Assume that q is not a root of unity. Then the algebra Z (∆) is
generated by Ω, α, β, γ.
Moreover Z (∆) is isomorphic to a polynomial algebra in four
variables.
Paul Terwilliger
The universal Askey-Wilson algebra
The algebra ∆ and Uq (sl2 )
Our next goal is to explain how ∆ is related to the quantum group
Uq (sl2 ).
Definition
The F-algebra U = Uq (sl2 ) is defined by generators e, f , k ±1 and
relations
kk −1 = k −1 k = 1,
kf = q −2 fk,
k − k −1
ef − fe =
.
q − q −1
ke = q 2 ek,
We call e, f , k ±1 the Chevalley generators for U.
Paul Terwilliger
The universal Askey-Wilson algebra
Irreducible modules for Uq (sl2 )
We review the finite-dimensional irreducible modules for Uq (sl2 ).
Lemma
For all integers d ≥ 0 and ε ∈ {1, −1} there exists a U-module
Vd,ε with the following property: Vd,ε has a basis {vi }di=0 such that
kvi
= εq d−2i vi
fvi
= [i + 1]q vi+1
evi
= ε[d − i + 1]q vi−1
(0 ≤ i ≤ d),
(0 ≤ i ≤ d − 1),
(1 ≤ i ≤ d),
fvd = 0,
ev0 = 0.
The U-module Vd,ε is irreducible provided that q is not a root of
unity.
Paul Terwilliger
The universal Askey-Wilson algebra
The Casimir element of Uq (sl2 )
Earlier we gave a Casimir element for ∆. The algebra U also has a
Casimir element, which we now recall.
Definition
Define Λ ∈ U as follows:
Φ = ef (q − q −1 )2 + q −1 k + qk −1 .
We call Λ the (normalized) Casimir element of U.
Paul Terwilliger
The universal Askey-Wilson algebra
The Casimir element of Uq (sl2 ), cont.
The following result is well known.
Lemma
The Casimir element Λ is in the center Z (U). Moreover on the
U-module Vd,ε
Λ = ε(q d+1 + q −d−1 )I .
Paul Terwilliger
The universal Askey-Wilson algebra
The equitable presentation of Uq (sl2 )
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
yy −1 = y −1 y
qxy − q −1 yx
q − q −1
qyz − q −1 zy
q − q −1
qzx − q −1 xz
q − q −1
= 1,
= 1,
= 1,
= 1.
We call x, y ±1 , z the equitable generators for U.
Paul Terwilliger
The universal Askey-Wilson algebra
The U-module Vd,ε from the equitable point of view
In the equitable presentation the U-module Vd,ε looks as follows.
Vd,ε has three bases such that:
basis 1
basis 2
basis 3
x
diagonal
upper bidiagonal
lower bidiagonal
y
lower bidiagonal
diagonal
upper bidiagonal
Paul Terwilliger
z
upper bidiagonal
lower bidiagonal
diagonal
The universal Askey-Wilson algebra
The Casimir element Λ of Uq (sl2 )
In the equitable presentation of U the Casimir element looks as
follows.
Lemma (Ter 2011)
The Casimir element Λ is equal to each of the following:
qx + q −1 y + qz − qxyz,
q −1 x + qy + q −1 z − q −1 zyx,
qy + q −1 z + qx − qyzx,
q −1 y + qz + q −1 x − q −1 xzy ,
qz + q −1 x + qy − qzxy ,
q −1 z + qx + q −1 y − q −1 yxz.
Paul Terwilliger
The universal Askey-Wilson algebra
∆ and Uq (sl2 )
We are now ready to describe how ∆ is related to Uq (sl2 ).
Lemma (Ter 2011)
Let a, b, c denote nonzero scalars in F. Then there exists an
F-algebra homomorpism ∆ → Uq (sl2 ) that sends
xy − yx −1
bc ,
q − q −1
yz − zy −1
ca ,
B 7→ yb + zb −1 +
q − q −1
zx − xz −1
C →
7
zc + xc −1 +
ab .
q − q −1
A 7→ xa + ya−1 +
The above homomorphism is not injective. To shrink the kernel we
do the following.
Paul Terwilliger
The universal Askey-Wilson algebra
∆ and Uq (sl2 )
From now on let a, b, c denote mutually commuting
indeterminates.
Let F[a±1 , b ±1 , c ±1 ] denote the F-algebra of Laurent polynomials
in a, b, c that have all coefficients in F.
Consider the F-algebra
U ⊗ F[a±1 , b ±1 , c ±1 ],
where U = Uq (sl2 ) and ⊗ = ⊗F .
Paul Terwilliger
The universal Askey-Wilson algebra
∆ and Uq (sl2 )
Theorem (Ter 2011)
There exists an injective F-algebra homomorphism
\ : ∆ → U ⊗ F[a±1 , b ±1 , c ±1 ] that sends
xy − yx
⊗ bc −1 ,
q − q −1
yz − zy
B 7→ y ⊗ b + z ⊗ b −1 +
⊗ ca−1 ,
q − q −1
zx − xz
C →
7
z ⊗ c + x ⊗ c −1 +
⊗ ab −1 ,
q − q −1
A 7→ x ⊗ a + y ⊗ a−1 +
where x, y , z denote the equitable generators for U.
Paul Terwilliger
The universal Askey-Wilson algebra
∆ and Uq (sl2 )
Theorem (Ter 2011)
The homomorphism \ sends
α 7→ Λ ⊗ (a + a−1 ) + 1 ⊗ (b + b −1 )(c + c −1 ),
β 7→ Λ ⊗ (b + b −1 ) + 1 ⊗ (c + c −1 )(a + a−1 ),
γ 7→ Λ ⊗ (c + c −1 ) + 1 ⊗ (a + a−1 )(b + b −1 ),
where Λ denotes the Casimir element of U.
Paul Terwilliger
The universal Askey-Wilson algebra
∆ and Uq (sl2 )
Theorem (Ter 2011)
Under the homomorphism \ the image of the Casimir element Ω is
1 ⊗ (q + q −1 )2 − Λ2 ⊗ 1 − 1 ⊗ (a + a−1 )2 − 1 ⊗ (b + b −1 )2
−1 ⊗ (c + c −1 )2 − Λ ⊗ (a + a−1 )(b + b −1 )(c + c −1 )
where Λ denotes the Casimir element of U.
Paul Terwilliger
The universal Askey-Wilson algebra
∆ and the universal DAHA of type (C1∨ , C1 ).
Our next goal is to describe how ∆ is related to the double affine
Hecke algebra (DAHA) of type (C1∨ , C1 ).
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
I = {0, 1, 2, 3}.
Paul Terwilliger
The universal Askey-Wilson algebra
The universal DAHA of type (C1∨ , C1 )
Definition
Let Ĥq denote the F-algebra defined by generators {ti±1 }i∈I and
relations
ti ti−1 = ti−1 ti = 1
ti +
ti−1
i ∈ I,
i ∈ I,
is central
t0 t1 t2 t3 = q
−1
.
We call Ĥq the universal DAHA of type (C1∨ , C1 ).
For notational convenience define
Ti = ti + ti−1
Paul Terwilliger
i ∈ I.
The universal Askey-Wilson algebra
The elements X , Y of Ĥq
We will describe how ∆ is related to Ĥq .
To set the stage we first mention a few basic features of Ĥq .
Define
X = t3 t0 ,
Y = t0 t1 .
Note that X , Y are invertible.
Paul Terwilliger
The universal Askey-Wilson algebra
A basis for Ĥq
Theorem (Ter 2012)
The following is a basis for the F-vector space Ĥq :
Y i X j t0k T1r T2s T3t
i, j, k ∈ Z
r , s, t ∈ N.
This can be proven using the Bergman Diamond Lemma.
Paul Terwilliger
The universal Askey-Wilson algebra
A basis for Ĥq
Corollary (Ter 2012)
The following are algebraically independent over F:
t0 ,
T1 ,
T2 ,
T3 .
Corollary (Ter 2012)
The following are algebraically independent over F:
T0 ,
T1 ,
Paul Terwilliger
T2 ,
T3 .
The universal Askey-Wilson algebra
The center Z (Ĥq )
We now describe the center Z (Ĥq ).
Theorem (Ter 2012)
Assume that q is not a root of unity. Then the algebra Z (Ĥq ) is
generated by {Ti }i∈I .
Moreover Z (Ĥq ) is isomorphic to a polynomial algebra in four
variables.
Paul Terwilliger
The universal Askey-Wilson algebra
Some automorphisms of Ĥq
We mention some automorphisms of Ĥq .
We start with an obvious one.
There exists an automorphism of Ĥq that sends
t0 7→ t1 7→ t2 7→ t3 7→ t0 .
We call this Z4 -symmetry.
This symmetry sends
X 7→ Y 7→ q −1 X −1 7→ q −1 Y −1 7→ X .
Paul Terwilliger
The universal Askey-Wilson algebra
The Artin braid group B3
We will be discussing the Artin braid group B3 .
Definition
The group B3 is defined by generators ρ, σ and relations ρ3 = σ 2 .
For notational convenience define τ = ρ3 = σ 2 .
There exists a group homomorphism B3 → PSL2 (Z) that sends
ρ 7→ p and σ 7→ s. Via this homomorphism we pull back the
PSL2 (Z) action on ∆, to get a B3 action on ∆ as a group of
automorphisms.
Next we explain how B3 acts on Ĥq as a group of automorphisms.
Paul Terwilliger
The universal Askey-Wilson algebra
An action of B3 on Ĥq
Lemma
The group B3 acts on Ĥq as a group of automorphisms such that
τ (h) = t0−1 ht0 for all h ∈ Ĥq and ρ, σ do the following:
h
ρ(h)
σ(h)
t0
t0
t0
t1
−1
t0 t3 t0
t0−1 t3 t0
Paul Terwilliger
t2
t1
t1 t2 t1−1
t3
t2
t1
The universal Askey-Wilson algebra
An action of B3 on Ĥq , cont.
Lemma
The B3 action on Ĥq does the following to the central elements
{Ti }i∈I . The generator τ fixes every central element. The
generators ρ, σ satisfy the table below.
h
ρ(h)
σ(h)
T0
T0
T0
T1
T3
T3
T2
T1
T2
T3
T2
T1
We are now ready to describe how ∆ is related to Ĥq .
Paul Terwilliger
The universal Askey-Wilson algebra
∆ and Ĥq
Theorem (Ter 2012)
There exists an injective F-algebra homomorphism ψ : ∆ → Ĥq
that sends
A 7→ t1 t0 + (t1 t0 )−1 ,
B 7→ t3 t0 + (t3 t0 )−1 ,
C
7→ t2 t0 + (t2 t0 )−1 .
Paul Terwilliger
The universal Askey-Wilson algebra
∆ and Ĥq
Theorem (Ter 2012)
The homomorphism ψ sends
α 7→ (q −1 t0 + qt0−1 )(t1 + t1−1 ) + (t2 + t2−1 )(t3 + t3−1 ),
β 7→ (q −1 t0 + qt0−1 )(t3 + t3−1 ) + (t1 + t1−1 )(t2 + t2−1 ),
γ 7→ (q −1 t0 + qt0−1 )(t2 + t2−1 ) + (t3 + t3−1 )(t1 + t1−1 ).
Paul Terwilliger
The universal Askey-Wilson algebra
∆ and Ĥq
Theorem (Ter 2012)
Under the homomorphism ψ the image of the Casimir element Ω is
(q + q −1 )2 − (q −1 t0 + qt0−1 )2 − (t1 + t1−1 )2 − (t2 + t2−1 )2
− (t3 + t3−1 )2 − (q −1 t0 + qt0−1 )(t1 + t1−1 )(t2 + t2−1 )(t3 + t3−1 ).
Paul Terwilliger
The universal Askey-Wilson algebra
∆ and Ĥq
Theorem (Ter 2012)
For all g ∈ B3 the following diagram commutes:
ψ
∆ −−−−→


gy
Ĥq

g
y
∆ −−−−→ Ĥq
ψ
Paul Terwilliger
The universal Askey-Wilson algebra
The image of ∆ under ψ
Now consider the image of ∆ under ψ.
As we will see, this image is related to the “spherical subalgebra”
{h ∈ Ĥq |t0 h = ht0 }.
Paul Terwilliger
The universal Askey-Wilson algebra
The image of ∆ under ψ
Theorem (Ter 2012)
Consider the image of ∆ under ψ. The spherical subalgebra
{h ∈ Ĥq |t0 h = ht0 } is generated by this image together with
t0±1 , T1 , T2 , T3 .
Paul Terwilliger
The universal Askey-Wilson algebra
Notation
For notational convenience, from now on identify ∆ with its image
under the injection ψ : ∆ → Ĥq .
From this point of view
A = t1 t0 + (t1 t0 )−1 = t0 t1 + (t0 t1 )−1 = Y + Y −1 ,
B = t3 t0 + (t3 t0 )−1 = t0 t3 + (t0 t3 )−1 = X + X −1 ,
C = t2 t0 + (t2 t0 )−1 = t0 t2 + (t0 t2 )−1 ,
α = (q −1 t0 + qt0−1 )T1 + T2 T3 ,
β = (q −1 t0 + qt0−1 )T3 + T1 T2 ,
γ = (q −1 t0 + qt0−1 )T2 + T3 T1 ,
Ω = (q + q −1 )2 − (q −1 t0 + qt0−1 )2 − T12 − T22 − T32
− (q −1 t0 + qt0−1 )T1 T2 T3 .
Paul Terwilliger
The universal Askey-Wilson algebra
A presentation for the spherical subalgebra by generators
and relations
We now give a presentation of the spherical subalgebra
{h ∈ Ĥq |t0 h = ht0 } by generators and relations.
This will be our last result of the talk.
Paul Terwilliger
The universal Askey-Wilson algebra
A presentation for the spherical subalgebra
Theorem (Ter 2012)
The spherical subalgebra {h ∈ Ĥq |t0 h = ht0 } is presented by
generators and relations in the following way. The generators are
A, B, C , t0±1 , {Ti }3i=1 . The relations assert that each of t0±1 ,
{Ti }3i=1 is central and t0 t0−1 = 1, t0−1 t0 = 1,
α
qBC − q −1 CB
=
,
2
−2
q −q
q + q −1
β
qCA − q −1 AC
=
,
B+
q 2 − q −2
q + q −1
qAB − q −1 BA
γ
C+
=
,
2
−2
q −q
q + q −1
q −1 ACB + q −2 A2 + q −2 B 2 + q 2 C 2 − q −1 Aα − q −1 Bβ − qC γ
A+
= (q + q −1 )2 − (q −1 t0 + qt0−1 )2 − T12 − T22 − T32
−(q −1 t0 + qt0−1 )T1 T2 T3 ,
Paul Terwilliger
The universal Askey-Wilson algebra
A presentation for the spherical subalgebra, cont.
Theorem
where
α = (q −1 t0 + qt0−1 )T1 + T2 T3 ,
β = (q −1 t0 + qt0−1 )T3 + T1 T2 ,
γ = (q −1 t0 + qt0−1 )T2 + T3 T1 .
Paul Terwilliger
The universal Askey-Wilson algebra
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 PSL2 (Z) acts on ∆ as a group of automorphisms.
We described how ∆ is related to Uq (sl2 ).
Finally we described how ∆ is related to the universal DAHA of
type (C1∨ , C1 ).
Thank you for your attention!
THE END
Paul Terwilliger
The universal Askey-Wilson algebra
References
Hau-wen Huang. The classification of Leonard triples of QRacah
type. Linear Algebra Appl. To appear. arXiv:1108.0458.
T. Ito and P. Terwilliger. Double affine Hecke algebras of rank 1
and the Z3 -symmetric Askey-Wilson relations. SIGMA 6 (2010)
065, 9 pages, arXiv:1001.2764.
P. Terwilliger. The universal Askey-Wilson algebra. SIGMA 7
(2011) 069, 24 pages, arXiv:1104.2813.
P. Terwilliger. The universal Askey-Wilson algebra and the
equitable presentation of Uq (sl2 ). SIGMA 7 (2011) 099, 26 pages,
arXiv:1107.3544.
P. Terwilliger. The universal Askey-Wilson algebra and DAHA of
type (C1∨ , C1 ). Submitted for publication.
Paul Terwilliger
The universal Askey-Wilson algebra