Representations of Askey--Wilson algebra
Daniel Gromada Severin Pošta
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Representations of Askey–Wilson algebra

Daniel Gromada^†, Severin Pošta^


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^† Department of Physics, Faculty of Nuclear Sciences and Physical Engineering,

^† Czech Technical University in Prague, Břehová 7, CZ-115 19 Prague, Czech Republic

^† E-mail: d.gromada@seznam.cz

^ Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering,

^ Czech Technical University in Prague, Trojanova 13, CZ-120 00 Prague, Czech Republic

^ E-mail: severin.posta@fjfi.cvut.cz


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Abstract.We deal with the classification problem of finite-dimensional representations of so called Askey–Wilson algebra in the case when q is not a root of unity. We classify all representations satisfying certain property, which ensures diagonalizability of one of the generating elements.

Keywords: algebra, representation, orthogonal polynomials


Introduction

The relationship between special classes of orthogonal polynomials from the Askey scheme is already known for some time. Such a relationship is very valuable since the basic properties of the orthogonal polynomials can be derived from this relationship very easily. In 1991 <cit.>lexei Zhedanov constructed a q-commutator algebra that correspond to the most general class of discrete polynomials from the Askey–Wilson scheme—the q-Racah polynomials defined by Askey and Wilson in <cit.>see <cit.>or detailed overview of the Askey–Wilson scheme). Zhedanov named the algebra after authors of the scheme Richard Askey and James Wilson.

The important concept that emerges in both Askey–Wilson algebra and finite orthogonal polynomial sequences and connects those structures is so called Leonard pair, which is a pair of operators such that both are tridiagonal in the eigenbasis of the other. This correspondence was first discovered by Leonard <cit.> The pair of such operators was named after him by Terwilliger <cit.> For a nice introduction to the theory of Leonard pairs, see <cit.> The Leonard pair corresponding to q-Racah polynomials is studied in <cit.>

In the Askey–Wilson algebra the Leonard pair is made of elements of the algebra in certain representation. If we want to find the Leonard pair and discover the orthogonal polynomials and their properties without using the properties of the orthogonal polynomials in the first place, we can make use of classification of representations and certain automorphisms, as we have shown for example in <cit.> Although Zhedanov was able to construct suitable representation of the Askey–Wilson algebra and show the connection with q-Racah polynomials, the complete classification was not available until recent work of Hau-Wen Huang <cit.> (Only in the case when q is not a root of unity. The case when q is a root of unity is much more complicated and the classification problem is still open, see <cit.>)

Independently on Huang we worked on the classification problem for Askey–Wilson algebra as well. Our approach follows the Zhedanov's paper <cit.>sing the technique of shift operators to construct the representations. However, it completes the Zhedanov's paper with rigorous mathematical theorems and proofs. The technique is inspired by classification of representations of algebra U'_q(_3) <cit.> which is a special case of Askey–Wilson algebra.

We show that every representation satisfying certain property (certain numbers should not be eigenvalues of one of the generating element) can be constructed using shift operators and that those representations are determined by trace of the generating element (and parameters of the algebra of course). Consequently, we were able to prove that the dual representation constructed by Zhedanov, which provides the relationship with q-Racah polynomials, is indeed equivalent to the original one without need of prior knowledge of the properties of the q-Racah polynomials.

We also consider separately special case when certain parameters of the Askey–Wilson algebra are zero, which allows to construct new representations by the shift operators not considered by Zhedanov.

Although complete classification of Askey–Wilson algebra representations was already presented, we think that our approach is still interesting since it is very straightforward using the classical technique of shift operators and extends the Zhedanov's paper.

In the whole work, we assume that q is not a root of unity.

Definition and basic properties

The definition of Askey–Wilson algebra was presented by Alexey Zhedanov in <cit.> We will use a _3-symmetric presentation, which was mentioned for example in <cit.> We chose the presentation in a way that the algebra U'_q(_3) is a special case when all parameters equal to zero.

D.AW]
Askey–Wilson algebra _q(A_1,A_2,A_3) is a complex associative algebra generated by three elements I_1, I_2, I_3 and relations

    eq.AW1]	q^1/2I_1I_2-q^-1/2I_2I_1   =I_3+A_3,   eq.AW2]	q^1/2I_2I_3-q^-1/2I_3I_2   =I_1+A_1,   eq.AW3]	q^1/2I_3I_1-q^-1/2I_1I_3   =I_2+A_2,

where A_1, A_2 and A_3 are complex parameters. We will often denote the algebra only shortly as .

Later, Paul Terwilliger in <cit.>resented so called universal Askey–Wilson algebra, that was also defined by three generating elements satisfying the same commutation relations, but A_1, A_2 and A_3 are not considered as complex parameters, but as elements of the center of the algebra.










The set of monomials {I_1^kI_2^mI_3^n| k,m,n∈ N_0} forms a basis of .
As in the case of U'_q(_3) <cit.>r universal Askey–Wilson algebra <cit.>e make use of the Diamond lemma <cit.> We transform the generating relation into a form compatible with ordering I_1≤ I_2≤ I_3:

    I_2I_1   =qI_1I_2-q^1/2(I_3+A_3),
    I_3I_2   =qI_2I_3-q^1/2(I_1+A_1),
    I_3I_1   =q^-1I_1I_3+q^-1/2(I_2+A_2).

We see that there are no inclusion ambiguities on the left hand side and there is only one overlap ambiguity—a monomial I_3I_2I_1 can be reduced using the first or the second relation. We show that this ambiguity is resolvable.

Reducing by the first relation we get

    I_3I_2I_1   =qI_3I_1I_2-q^1/2I_3(I_3+A_3)=I_1I_3I_2+q^1/2(I_2(I_2+A_2)-I_3(I_3+A_3))   =qI_1I_2I_3+q^1/2(-I_1(I_1+A_1)+I_2(I_2+A_2)-I_3(I_3+A_3)),while the second relation leads to   =qI_2I_3I_1-q^1/2I_3(I_1+A_1)=I_2I_1I_3+q^1/2(-I_1(I_1+A_1)+I_2(I_2+A_2))   =qI_1I_2I_3+q^1/2(-I_1(I_1+A_1)+I_2(I_2+A_2)-I_3(I_3+A_3)).

Both results are the same, so the ambiguity is resolvable.

Finally, we can see that {I_1^kI_2^mI_3^n} is indeed the set of all reduced monomials.

L.AWiso]
There are the following isomorphisms of Askey–Wilson algebras

    ρ_q(A_1,A_2,A_3)→_q(A_2,A_3,A_1),   I_1↦ I_2,   I_2↦ I_3,   I_3↦ I_1,


    σ_q(A_1,A_2,A_3)→_q(A_2,A_1,A_3),


    I_1↦ I_2,   I_2↦ I_1,   I_3↦ I_3+(I_2I_1-I_1I_2)(q^1/2+q^-1/2),


    τ_ϵ,ϵ'_q(A_1,A_2,A_3)→_q(ϵ A_1,ϵ' A_2,ϵϵ' A_3),   I_1↦ϵ I_1,   I_2↦ϵ' I_2,   I_3↦ϵϵ' I_3,

where ϵ, ϵ'∈{1,-1}.

Those isomorphisms can be also interpreted as automorphisms of the universal Askey–Wilson algebra. In fact, the first two form a faithful action of the group PSL_2( Z) <cit.>

We are going to use the following definition of q-numbers

    [α]_q:=q^α-q^-α q-q^-1  for α∈.


L.qnum]
(Lemma 10 <cit.> Let q is not a root of unity.
n
	* If q^2α=-q^l for some l∈ Z, then [α-j]_q=[α-k]_q for j,k∈ Z, j≠ k, if and only if j+k=l.
	* Conversely, having [α-j]_q=[α-k]_q for some α∈ a j,k∈ Z, j≠ k, it follows that q^2α=-q^j+k.
	* For every λ∈ there exists α∈ such that λ=[α]_q and numbers [α]_q, [α+1]_q,[α+2]_q,… are mutually different.


Classification of finite-dimensional representations

As we indicated in the introduction, we follow the Zhedanov's construction of the representations. So, the definitions of Casimir element, shift operators, characteristic polynomial or dual representations are all inspired by the original paper <cit.>

L.casimir]
Casimir element
eq.casimir]

    C:=q^2I_1^2+I_2^2+q^2I_3^2-(q^5/2-q^1/2)I_1I_2I_3+q(q+1)A_1I_1+(q+1)A_2I_2+q(q+1)A_3I_3

is a central element of .

For suitable complex numbers λ, let O_λ and R_λ be the following linear combinations:

    eq.Odef]	O_λ   :=- q^1/2A_1+q^-λA_2 [λ]_q([λ]_q-[λ+1]_q)I_3+ I_2+q^-λ+1/2I_1,   eq.Rdef]	R_λ   :=- q^1/2A_1-q^λA_2 [λ]_q([λ]_q-[λ-1]_q)I_3+ I_2-q^λ+1/2I_1.

In case of A_1=A_2=0 we consider this definition without the first term and we assume that λ is an arbitrary complex number. Otherwise, we require the expression to be well defined, so [λ]_q≠[λ+1]_q and [λ]_q≠[λ-1]_q, respectively.


In the following text, considering a representation R of the algebra  on a vector space V, we will denote the representing linear operators just I_1, I_2, and I_3 instead of R(I_1), R(I_2), and R(I_3).

Let R be a representation of  on V. Let x∈(I_3+ [λ]_q). Then

    eq.Oshift]	I_3(O_λ x)   =- [λ+1]_qO_λ x,   eq.Rshift]	I_3(R_λ x)   =- [λ-1]_qR_λ x,   eq.OR]	O_λ-1R_λ x   =(C̃_λ-1-C)x,   eq.RO]	R_λ+1O_λ x   =(C̃_λ-C)x,

where

    C̃_λ=-q[λ]_q[λ+1]_q- q([λ]_q+[λ+1]_q)A_3-qA_1^2+ (q^-λ-1/2-q^λ+1/2)A_1A_2+A_2^2([λ]_q-[λ+1]_q)^2.

By inspection using the q-commutation relations <ref>eq.AW1]–<ref>eq.AW3].

L.I1I2soust]
Take y,z∈ V and λ∈ such that both O_λ and R_λ are well-defined. Then the system of linear equations

    O_λ x=y,   R_λ x=z

for vectors I_1x and I_2x has a unique solution if and only if q^λ≠ϵ for ϵ∈{-1,1}.

From <ref>eq.Odef] and <ref>eq.Rdef] we can easily express (q^-λ+q^λ)I_1x. The factor in the bracket is nonzero if and only if q^λ≠ϵ. The same holds for I_2x.

Now, consider a general representation of Askey–Wilson algebra. We construct a basis in which I_3 acts diagonally and we try to express the explicit form of such representation. The construction of the basis is an analogy of Theorem 3 from <cit.> where the same procedure is performed for U'_q(_3). To perform such a construction, we need to assume that 2ϵ/(q-q^-1) and ϵ/(q^1/2-q^-1/2), ϵ=± 1 are not an eigenvalues of I_3. Such representations will be called classical (inspired by notation in <cit.>.

These conditions allow us to use the shift operators to construct the eigenbasis of I_3. The first condition ensures the existence of solution of the system in lemma <ref>L.I1I2soust] and the second one ensures the shift operators to be well defined.
However, in the case when A_1=A_2=0, which will be considered separately, we will be able to construct the eigenbasis of I_3 even for the non-classical representations.






secc.obrep]
Classical representations

V.AWbaze]
Let q is not a root of unity. Let R be an irreducible classical representation (i.e. satisfying  (I_3-2ϵ/(q-q^-1))={0} and (I_3-ϵ/(q^1/2-q^-1/2))≠{0} for ϵ=± 1) of _q(A_1,A_2,A_3) on V, V=N+1. Then there exists a complex number μ and a non-zero vector v_0∈(I_3+[μ]_q) such that O_μ v_0=0. If we define

    v_j+1:=R_μ-jv_j  for j=0,1,… r-1,

then the tuple (v_0,…,v_N) forms a basis of V.
Let w_0 be an eigenvector corresponding to eigenvalue -[μ̃]_q, where μ̃ is chosen in such a way that the numbers -[μ̃]_q, -[μ̃+1]_q,… are mutually different (see Lemma <ref>L.qnum]). Now, define

    w_j+1:=O_μ̃+jw_j  for j≥ 0

and denote l∈ such that w_0,…,w_l-1 are linearly independent, but w_l is already linearly dependent on them. Such l has to exist since V is finite-dimensional. From the equation <ref>eq.Oshift] it follows that the vectors w_j satisfy the eigenequation I_3w_j=-[μ̃+j]_qw_j, where all the eigenvalues are mutually different. Since eigenvectors corresponding to mutually different eigenvalues are linearly independent, we have w_l=0.

Denote v_0:=w_l-1 a μ:=μ̃+l-1 and define the following vectors:
eq.vjdef]

    v_j+1:=R_μ-jv_j  for j≥ 0.

From the equation <ref>eq.Rshift] it follows that they also have to satisfy the eigenequations
eq.I3v]

    I_3v_j=-[μ-j]_qv_j.

The possibility of [μ-j]_q=[μ-j-1]_q for some j, so R_μ-j would not be well defined, contradicts the assumptions since it would follow that q^μ=ϵ q^j-1/2, so -[μ-j]_q=ϵ/(q^1/2-q^-1/2) would be an eigenvalue of I_3. Now we can denote k∈ such that the tuple v_0,…,v_k-1 is linearly independent, while v_k is already linearly dependent. From <ref>eq.OR] it follows that
eq.Ov]

    O_μ-jv_j=O_μ-jR_μ-j+1v_j-1=(C̃_μ-j-C)v_j-1,   O_μ v_0=y_l=0

for j=1,2,…,k-1.

From the irreducibility of the representation it follows that C is a multiple of identity. This complex number will be also denoted by C. Its value is determined by equality 0=R_μ+1O_μ v_0=(C̃_μ-C)v_0, so C=C̃_μ.

The equations <ref>eq.vjdef] and <ref>eq.Ov] define a system of equations
eq.I1I2soust]

    R_μ-jv_j=v_j+1,   O_μ-jv_j=(C̃_μ-j-C)v_j-1,

which, according to Lemma <ref>L.I1I2soust], has a solution for I_1v_j a I_2v_j if and only if q^μ-j≠ϵ for ϵ∈{-1,1}. This condition is satisfied thanks to our assumptions, otherwise [μ-j]_q=2ϵ/(q-q^-1) would be an eigenvalue of I_3 corresponding to the eigenvector v_j. Therefore, we were able to express the vectors I_1v_j and I_2v_j as a linear combination of v_0,…,v_k-1. Moreover, I_3 acts diagonally, so the span of {v_0,…,v_k-1} is an invariant subspace and thanks to the irreducibility it has to be equal to the whole space V.


L.klasvlč]
For arbitrary (N+1)-dimensional classical irreducible representation the operator I_3 is diagonalizable and has mutually different eigenvalues. Its eigenvalues are -[μ]_q, -[μ-1]_q, …, -[μ-N]_q. Denoting v_0, …, v_N the corresponding eigenvectors, we have O_μ v_0=0 and v_r:=R_μ-Nv_N=0. In addition, it holds that q^2μ≠ q^l for every l∈{-1,0,1,…,2N+1}.
The diagonalizability and spectrum of I_3 follows from the previous theorem. From assumptions, we have -[μ-j]_q≠ 2ϵ/(q-q^-1), so q^μ≠ϵ q^j for j=0,…,N. We also assume that -[μ-j]_q≠ϵ/(q^1/2-q^-1/2), which means q^μ≠ϵ q^j± 1/2 for j=0,…,N. Together it follows that q^μ≠ϵ q^l/2, l=-1,…,2N+1.

This condition implies that the numbers [μ+1]_q, …, [μ-N-1]_q are mutually different according to Lemma <ref>L.qnum].

The vector O_μ v_0 is either zero or an eigenvector corresponding to eigenvalue -[μ+1]_q. The second possibility cannot take place since -[μ+1]_q is not in the spectrum of I_3. The same holds for the vector R_μ-Nv_N.

In the following text, we will still consider a representation satisfying the assumptions of Theorem <ref>V.AWbaze].

We will often deal with sums and products of consecutive q-numbers. Hence, it will be useful to summarize following relations

    ([λ+1]_q±[λ]_q)(q^1/2∓ q^-1/2)=q^λ+1/2∓ q^-λ-1/2,


    [λ]_q[λ+1]_q=([λ]_q+[λ+1]_q)^2-1 (q^1/2+q^-1/2)^2,


    ([λ]_q-[λ+1]_q)^2=([λ]_q+[λ+1]_q)^2(q^1/2-q^-1/2)^2(q^1/2+q^-1/2)^2+4(q^1/2+q^-1/2)^2.


We denote D̃_j:=C̃_μ-C̃_μ-j which will play an important role for characterization of the representations. Next, we denote
eq.lambdadef]

    Λ_j:=[μ-j+1]_q+[μ-j]_q=q^μ-j+1/2-q^-μ+j-1/2(q^1/2-q^-1/2),


    g_j:=[μ-j+1]_q-[μ-j]_q=q^μ-j+1/2+q^-μ+j-1/2(q^1/2+q^-1/2).

Using the relations above and the definition of C̃_μ we see that D̃_jg_j^2 is a polynomial of fourth order in variable Λ_j. The roots of this polynomial determine the form of the representation. Thus, we could call it a characteristic polynomial. (Characteristic polynomial is defined in <cit.>n a similar way.) From the definition of D̃_j it is clear that D̃_0=0; thus, one of the roots of the characteristic polynomial is Λ_0.

Solving the system <ref>eq.I1I2soust] we get
eq.I1v]

    I_1v_j=-q^-1/2 q^-μ+j+q^μ-j(D̃_jv_j-1+v_j+1)-A_1- [μ-j]_qA_2(q^1/2-q^-1/2) g_jg_j+1v_j,

eq.I2v]

    I_2v_j= q^-μ+j+q^μ-j(q^μ-jD̃_jv_j-1-q^-μ+jv_j+1)-- [μ-j]_qA_1(q^1/2-q^-1/2)+A_2 g_jg_j+1v_j.


Denote Λ_j_0,…,Λ_j_3 the roots of the characteristic polynomial and choose j_0=0. Since we know, what is the leading coefficient, we can factor the polynomial
eq.Djjk]

    D̃_j   =q(q^1/2-q^-1/2)^2(q^1/2+q^-1/2)^4∏_k=0^3(Λ_j-Λ_j_k) g_j^2   =q∏_k=0^3(q^μ-j+1/2-q^-μ+j-1/2-q^μ-j_k+1/2+q^-μ+j_k-1/2)(q-q^-1)^2(q^μ-j+1/2+q^-μ+j-1/2)^2   =q^2μ-2j+2∏_k=0^3((1-q^j-j_k)(1+q^-2μ+j+j_k-1))(q-q^-1)^2(1+q^-2μ+2j-1)^2


L.chpolkor]
For any (N+1)-dimensional classical irreducible representation the numbers Λ_0 and Λ_r are mutually different roots of the characteristic polynomial. For any k∈{1,…,N} the number Λ_k is not a root of the characteristic polynomial.
We can see that if D̃_k=0 for k∈{1,…,N}, it would mean that the span of v_k,…,v_N is a non-trivial invariant subspace, which contradicts the irreducibility of the representations.

Substituting <ref>eq.I1v], <ref>eq.I2v] and <ref>eq.I3v] to the condition (q^1/2I_1I_2-q^-1/2I_2I_1-I_3-A_3)v_N=0 (from the relation <ref>eq.AW1]) we get the following relation
eq.Dcond]

    - q^-1D̃_N+1 q^μ-N+q^-μ+Nv_N=0,

which implies D̃_N+1=0, so Λ_N+1 is a root of the characteristic polynomial as well. Now, we have to show that it is not equal to Λ_0.

Assuming Λ_0=Λ_N+1 and using the fact that q is not a root of unity we get q^2μ=-q^N, which is excluded by Lemma <ref>L.klasvlč].


The form of the representation of course depends on the parameters of the algebra A_1, A_2, and A_3. Besides that, it also depends on the value of the Casimir element C, which is determined by the number q^μ. The possible values of q^μ are restricted by the assumption of finite dimension D̃_N+1=0. This is a polynomial equation of degree eight in variable q^μ. Excluding the possibility Λ_N+1=Λ_0, we can reduce the degree to six by eliminating the factor Λ_N+1-Λ_0. However, the equation remains very hard to solve and we will not try to express the possible values of q^μ explicitly.

Representations with different q^μ may be equivalent. We will show that for fixed parameters of the algebra, the class of (N+1)-dimensional equivalent representations is determined by number [μ-N/2]_q, which is, up to a constant, the trace of I_3 (traces were used to distinguish representations already in the case of U'_q(_3) <cit.>nd they are also used in the Huang's classification <cit.>.

L.Rekviv]
Let R_1 and R_2 be classical irreducible r-dimensional representations of  constructed in Theorem <ref>V.AWbaze], {v_j^(1)} and {v_j^(2)} the corresponding bases, and μ_1 and μ_2 the corresponding complex numbers characterizing the representations. Then the representations R_1 and R_2 are equivalent if and only if [μ_1-N/2]_q=[μ_2-N/2]_q.
We compute the trace of I_3
eq.tr]

    I_3=∑_j=0^N-[μ-j]_q=-q^(N+1)/2-q^-(N+1)/2 q^1/2-q^-1/2[μ-N/2]_q.

Therefore, representations with different [μ-N/2]_q have to be inequivalent.

Conversely, let [μ_1-N/2]_q=[μ_2-N/2]_q. The equation <ref>eq.tr] is quadratic in q^μ-N/2 so it can have two solutions. Generally, it holds that [λ]_q=-[-λ]_q. If q^μ_1-N/2 is one of the solutions, the second one has to be q^μ_2-N/2=-q^-μ_1+N/2. This implies that

    [μ_1-j]_q=[μ_2-N+j]_q,   q^μ_1-j+q^-μ_1+j=-(q^μ_2-N+j+q^-μ_2+N-j),  Λ^(1)_j=Λ^(2)_N+1-j,

where Λ^(1)_j a Λ^(2)_j correspond to the definition <ref>eq.lambdadef] for μ_1 and μ_2. Similarly, we denote D̃_j^(1) and D̃_j^(2) and we define a basis {w_0,…,w_r-1} by equation

    v_j^(2)=∏_k=1^jD̃_k^(1)· w_N-j.

Using <ref>eq.I1v], <ref>eq.I2v], and <ref>eq.I3v] we can check that the representation R_1 has the same matrix elements in the basis {v_j^(1)} as R_2 in {w_j}.


To demonstrate the relationship of the representations with orthogonal polynomials it will be convenient to express the representation in terms of the numbers j_1, j_2, and j_3 instead of the parameters of the algebra A_1, A_2, and A_3. The parameters of the algebra together with the parameter of the representation q^μ determine the form of D̃_j, and hence the numbers Λ_j_1, Λ_j_2 a Λ_j_3. 

Expressing the numbers q^j_1, q^j_2 a q^j_3 in terms of the parameters of the algebra would lead to a bicubic equation, which would be very complicated and we will not perform it. To express the form of the representation in terms of those numbers, we will need the opposite relationship—to express the parameters of the algebra. This can be achieved using Vièt formulas for polynomial D̃_j in the variable Λ_j. For given numbers μ, j_1, j_2, j_3 we obtain the parameters A_1, A_2, A_3 such that Λ_j_1, Λ_j_2 a Λ_j_3 are roots of D̃_j.

Using one of the relations, we are able to express
eq.A3jk]

    A_3= (q^1/2+q^-1/2)^2(Λ_0+Λ_j_1+Λ_j_2+Λ_j_3),

and using the others we express a system of equations for A_1 and A_2

    (q^1/2-q^-1/2)^3(Λ_0Λ_j_1Λ_j_2+Λ_j_1Λ_j_2Λ_j_3+Λ_j_2Λ_j_3Λ_0+Λ_j_3Λ_0Λ_j_1)-   4(q^1/2-q^-1/2)(Λ_0+Λ_j_1+Λ_j_2+Λ_j_3)=(q^1/2+q^-1/2)^2(q-q^-1)^2A_1A_2,


    -   4(q^1/2-q^-1/2)^2(Λ_0(Λ_j_1+Λ_j_2+Λ_j_3)+Λ_j_1(Λ_j_2+Λ_j_3)+Λ_j_2Λ_j_3)+   (q^1/2-q^-1/2)^4Λ_0Λ_j_1Λ_j_2Λ_j_3+16=(q^1/2+q^-1/2)^2(q-q^-1)^2(A_1^2+A_2^2).


This system leads to a biquadratic equation, so it has four solutions. From symmetry of the problem it is evident that if we denote (A_1,A_2) one of the solutions, the other solutions are (-A_1, -A_2), (A_2,A_1) a (-A_2,-A_1). The explicit form of one of the solutions is following

    eq.A1jk]
    A_1=   (q^1/2+q^-1/2)(q-q^-1)   (   q^(j_1+j_2+j_3-2μ-1)/2+q^(-j_1+j_2+j_3-2μ-1)/2+q^(j_1-j_2+j_3-2μ-1)/2+q^(j_1+j_2-j_3-2μ-1)/2
    -   q^(2μ+1-j_1-j_2+j_3)/2-q^(2μ+1-j_1-j_2-j_3)/2-q^(2μ+1+j_1-j_2-j_3)/2-q^(2μ+1-j_1+j_2-j_3)/2),eq.A2jk]
    A_2=   1(q^1/2+q^-1/2)(q-q^-1)      (q^-2μ-1+(j_1+j_2+j_3)/2-q^(-j_1+j_2+j_3)/2-q^(j_1-j_2+j_3)/2-q^(j_1+j_2-j_3)/2   -q^-(-j_1+j_2+j_3)/2-q^-(j_1-j_2+j_3)/2-q^-(j_1+j_2-j_3)/2+q^2μ+1-(j_1+j_2+j_3)/2).


We have found a representation for four isomorphic Askey–Wilson algebras with different parameters. If the necessary conditions for (N+1)-dimensional classical representation given by lemmata <ref>L.klasvlč] and <ref>L.chpolkor] are satisfied, then, by explicit computation, we can check that this representation really satisfies the commutation relations <ref>eq.AW1]–<ref>eq.AW3]. Since I_3 has mutually different eigenvalues, we can show the irreducibility of this representation easily. Taking an invariant subspace and an eigenvector of I_3 lying in this subspace. Using the shift operators O_λ a R_λ we construct the remaining elements of the eigenbasis. Therefore, we have proven the following theorem.

V.obecklas]
Let q is not a root of unity. For arbitrary classical irreducible representation of Askey–Wilson algebra there exist complex numbers μ, j_1, j_2, and j_3 such that this representation is equivalent to the representation given by equations <ref>eq.I3v], <ref>eq.I1v]–<ref>eq.Djjk]. Conversely, let the complex numbers μ, j_1, j_2, and j_3 satisfy the following assumptions: q^2μ≠- q^l for every l∈{-1,0,1,…,2N+1}, one of the numbers Λ_j_1, Λ_j_2, or Λ_j_3 equals to Λ_N+1, and none of those numbers equals to Λ_k for all k∈{1,…,N}. Then the equations <ref>eq.I3v], <ref>eq.I1v]–<ref>eq.Djjk] define an irreducible representations of algebras _q(A_1,A_2,A_3), _q(-A_1,-A_2,A_3), _q(A_2,A_1,A_3), and _q(-A_2,-A_1,A_3), where the numbers A_1, A_2, and A_3 are determined by equations <ref>eq.A3jk]–<ref>eq.A2jk].

P.assumptions]
The assumptions can be somehow reformulated and a bit simplified. Firstly, the representation obviously does not depend on the order of the numbers j_1, j_2, j_3, so we can fix, for example Λ_j_3=Λ_r. Note, however, that the q-Racah polynomials also depend on four parameters satisfying certain constraint and symmetry that could be used to eliminate one of them, so we will also keep all four parameters μ, j_1, j_2, and j_3. Secondly, the representation depend only on the roots Λ_j_1, Λ_j_2, and Λ_j_3, not on the numbers j_i nor q^j_i. So, we can fix j_3=r instead of Λ_j_3=Λ_r. Finally, it holds that Λ_j=Λ_k if and only if q^j=q^k or q^j=q^2μ-k+1, so the condition Λ_j_i≠Λ_r can be reformulated as q^j_i≠ q^r and q^j_i≠ q^2μ-r+1.


Our goal is to show the correspondence with orthogonal polynomials. For simplicity, we will work only with the solution <ref>eq.A1jk], <ref>eq.A2jk] and we will not express the form of I_2 in the following text. It could be computed easily in a similar way, but we will not need it.
eq.I1vobfin]

    I_1v_j=   -q^μ-j+j_1+j_2+j_3-1/21+q^-2μ+2j (q-q^-1)^2A_j-1C_jv_j-1      + q^-(2μ+2-j_1-j_2-j_3)/2 (q-q^-1)(A_j+C_j-1+q^2μ+2-j_1-j_2-j_3)v_j-q^-μ+j-1/2 1+q^-2μ+2jv_j+1,

where
eq.Aj]
    A_j=(1-q^j-j_1+1)(1-q^j-j_2+1)(1-q^j-j_3+1)(1+q^-2μ+j)(1+q^-2μ+2j)(1+q^-2μ+2j+1),

eq.Cj]
    C_j=-q^2μ+2-j_1-j_2-j_3(1-q^j)(1+q^-2μ-1+j+j_1)(1+q^-2μ-1+j+j_2)(1+q^-2μ-1+j+j_3)(1+q^-2μ+2j-1)(1+q^-2μ+2j).


secc.00]
Representations for A_1=A_2=0

V.AWneklbaze]
Let q is not a root of unity. Let R be an irreducible representation of _q(0,0,A_3) on V, V=N+1 satisfying (I_3-2ϵ/(q-q^-1))={0} for ϵ=± 1. Then there exists a complex number μ and a non-zero vector v_0∈(I_3+[μ]_q) such that O_μ v_0=0 and if we define

    v_j+1:=R_μ-jv_j  for j=0,1,… N,

then the tuple (v_0,…,v_N) forms a basis of V.
The construction can be performed in the same way as in Theorem <ref>V.AWbaze] (now, we do not have to ensure that [μ-j]_q≠[μ-j-1]_q).

The case A_1=A_2=0 is, of course, more similar to U'_q(_3), where all the parameters are zero. Here, the non-zero parameter A_3 causes shift of spectra of the representations.

Solving the system <ref>eq.I1I2soust] we get
eq.I1v00]
    I_1v_j=-q^-1/2 q^-μ+j+q^μ-j(D̃_jv_j-1+v_j+1),

eq.I2v00]
    I_2v_j= q^-μ+j+q^μ-j(q^μ-jD̃_jv_j-1-q^-μ+jv_j+1).


Here, we make use of the fact that D̃_j itself is a polynomial of degree two in Λ_j and factorize it. We get
eq.Dj00]

    D̃_j=q(q^1/2+q^-1/2)^2(Λ_j-Λ_0)(Λ_j+Λ_0+(q^1/2+q^-1/2)^2A_3).


Now, consider a classical representation, we will come back to the non-classical case later. We can, of course, use Lemma <ref>L.klasvlč] also in this particular case, so we have v_N+1=0 and D̃_N+1=0. This holds if Λ_N+1=Λ_0 or Λ_N+1=-Λ_0-(q^1/2+q^-1/2)^2A_3. The first possibility was already excluded in the general case.

So, assume Λ_N+1=-Λ_0-(q^1/2+q^-1/2)^2A_3. Rearranging the equality we get
eq.nyA3]

    (q^-(N+1)/2+q^(N+1)/2)[μ-N/2]_q=-(q^1/2+q^-1/2)A_3.

The bracket on the left-hand side cannot be zero since q is not a root of unity. Thus, it is a quadratic equation in q^μ-N/2. In Lemma <ref>L.Rekviv] we have already proven that the representations corresponding to those two solutions have to be equivalent.

Finally, we have to decide when the representation is irreducible.
In Lemma <ref>L.klasvlč] we have shown that in the classical case we have q^2μ≠ -q^l, where l∈{-1,0,1,…,2N+1} and the operator I_3 has mutually different eigenvalues. We can show the irreducibility in the same way as in the general case. The condition we mentioned can be rewritten in terms of the parameter A_3 by substituting into the equation <ref>eq.nyA3]:
eq.A3ner]

    (q^(N+1)/2+q^-(N+1)/2)(q^(k-N)/2+q^-(k-N)/2) q-q^-1≠ -ϵ(q^1/2+q^-1/2)A_3,   k∈{-1,0,1,…,2N+1}.


The final formulas for the representation can be expressed in the following form:
eq.I1vklas00]
    I_1v_j=   -q^μ-j+3/2 (1+q^-2μ+2j)(q-q^-1)(1-q^j)(1-q^j-N-1)   (1+q^-2μ+j-1)(1+q^-2μ+j+N)v_j-1+-q^-μ+j-1/2 1+q^-2μ+2jv_j+1,

eq.I2vklas00]
    I_2v_j=   - q^2μ-2j+2 (1+q^-2μ+2j)(q-q^-1)(1-q^j)(1-q^j-N-1)   (1+q^-2μ+j-1)(1+q^-2μ+j+N)v_j-1+- q^-2μ 1+q^-2μ+2jv_j+1,

eq.I3vklas00]
    I_3v_j=-q^μ-j-q^-μ+j q-q^-1v_j,

where μ is an arbitrary number satisfying <ref>eq.nyA3], j∈{0,…,N}, and x_-1=x_N+1=0. We can see that those representations correspond to the general ones analysed in the previous section for q^j_1=q^r, q^j_2=- q^μ+1/2, and q^j_3= q^μ+1/2 (the order is, of course, irrelevant).

Now, we move to the case of non-classical representations.

Consider the same notation as in Theorem <ref>V.AWneklbaze]. Let R be an (N+1)-dimensional non-classical irreducible representation of _q(0,0,A_3) such that (I_3-2ϵ/(q-q^-1))={0} for ϵ=± 1. Than the operator I_3 is diagonalizable and has mutually different eigenvalues, which correspond to eigenvectors v_0,…,v_N. In addition, we have [μ-N]_q=[μ-N-1]_q, i.e. q^μ=ϵ q^N+1/2 for ϵ∈{-1,1} and v_N+1=av_N, where
eq.a]

    a^2=-q[N+1]_q^2-ϵ q(q^(N+1)/2-q^-(N+1)/2)^2 q-q^-1A_3.

From the assumptions it follows that ϵ/(q^1/2-q^-1/2) is an eigenvalue of I_3, so there exists k∈{0,…,N} such that -[μ-k]_q=ϵ/(q^1/2-q^-1/2), so q^μ=ϵ q^k± 1/2, and so [μ-k]_q=[μ-(k± 1)]_q. Equivalently, there exists k∈{0,…,N+1} such that q^μ=ϵ q^k-1/2, i.e. [μ-k]_q=[μ-k+1]_q=ϵ/(q^1/2-q^-1/2). From the way of construction of the basis {v_j} it is clear that we cannot have [μ]_q=[μ+1]_q, so we have k>0. Next, we show that for k<N+1 we have a reducible representation. According to Lemma <ref>L.qnum] the equality k=N+1 means that the numbers [μ]_q,…,[μ-N]_q and hence the eigenvalues of I_3 are mutually different.

Consider q^μ=ϵ q^k-1/2, k∈{1,…,N+1}. According to Lemma <ref>L.qnum] we have [μ-N-1]_q=[μ-l]_q if and only if l=N+1 or l=2k-N. Therefore, the vector v_N+1 lies in an eigenspace corresponding to the eigenvalue -[μ-2k+N]_q. This subspace is one-dimensional for k>(N+1)/2, otherwise it is trivial. Thus, we can write v_N+1=av_2k-N, where v_j=0 for j≤ 0 and, for simplicity, we choose a=0 in this case, otherwise we have a∈.

Substituting <ref>eq.I1v00], <ref>eq.I2v00], and <ref>eq.I3v] into condition (q^1/2I_1I_2-q^-1/2I_2I_1-I_3-A_3)v_N (relation <ref>eq.AW1]) we get
eq.Dcond00nekl]

    D̃_N+1 q^N-k+1/2+q^-N+k-1/2v_N+a1 q^N-k+1/2-q^-N+k-1/2v_2k-N-1=0.

This is satisfied if a=0 or 2k-N-1=N+1. The equality 2k-N-1=N+1, i.e. k=N+1 means v_2k-N-1=v_N+1=av_N and q^μ=ϵ q^N+1/2. Substituting in <ref>eq.Dcond00nekl] we get a^2=-D̃_N+1. Substituting in <ref>eq.Dj00] we get the equality <ref>eq.a].

Now, consider k<N+1 and a=0, so k∈{1,…,N} and v_k=0, we show that the representation is reducible. According to Lemma <ref>L.qnum] we have [μ-j]_q=[μ-2k+j+1]_q for all j∈ Z. From the preceding equality it is easy to show that Λ_j=Λ_2k-j, so D̃_j=D̃_2k-j. Define

    w_j=∏_l=1^jD̃_l^-1/2· v_j+∏_l=1^2k-j-1D̃_l^-1/2· v_2k-j-1,

for j∈ Z, where an empty product is equal to one and v_j=0 for j∉{0,…,N}. It is a linear combination of vectors corresponding to the same eigenvalue (or zero vectors), so

    I_3w_j=-[μ-j]_qw_j.

We can also express

    I_1w_j=-ϵ q^-1/2 q^-k+j+1/2+q^k-j-1/2(D̃_j^1/2w_j-1+D̃_j+1^1/2w_j+1),


    I_2w_j=- q^-k+j+1/2-q^k-j-1/2(q^k-j-1/2D̃_j^1/2w_j-1+q^-k+j+1/2D̃_j+1^1/2w_j+1).

Since we have also w_j=w_2k-j-1 and, in particular, w_k-1=w_k and also w_-1=0 for k≥ (N+1)/2 or w_-N-1=0 for k≤ (N+1)/2, we can see that the span of the vectors w_k-1,…, w_0 for k≥ N or w_k,…,w_N for k≤ N forms an invariant subspace of the representation.


Since a is determined by <ref>eq.a] up to sign, we found additional four representations of the algebra. The proof of the irreducibility is the same as in the case of classical representations. The representations with different ϵ have different spectra of I_3. The representations with different a have different traces of I_1. Just in case of A_3=∓(q^(N+1)/2+q^-(N+1)/2)^2 and ϵ=± 1 we have a=0, so there are only three non-classical representations.

To write the final formulas we have to express

    D̃_j   =q(q^j/2-q^-j/2)(q^N+1-j/2-q^-N-1+j/2) q-q^-1((q^j/2+q^-j/2)(q^N+1-j/2+q^-N-1+j/2) q-q^-1+ϵ A_3)   =q[j]_q[2N+2-j]_q+qϵ(q^j/2-q^-j/2)(q^N+1-j/2-q^-N-1+j/2) q-q^-1A_3.

We are going to use the first row, but the second form illustrates the transition to the representations of U'_q(_3) (i.e. for A_3=0) listed in <cit.>
eq.I1vneklas00]
    I_1v_j=   ϵ (1-q^-2N+2j-1)(-q(1-q^j)(1-q^-2N-2+j) q-q^-1   (q^N+1-j(1+q^j)(1+q^-2N-2+j) q-q^-1+ϵ A_3)v_j-1+q^-N-1+jv_j+1),

eq.I2vneklas00]
    I_2v_j=    (1-q^-2N+2j-1)(-q^N-j+2(1-q^j)(1-q^-2N-2+j) q-q^-1   (q^N+1-j(1+q^j)(1+q^-2N-2+j) q-q^-1+ϵ A_3)v_j-1-q^-2N+2j-1v_j+1),

eq.I3vneklas00]
    I_3v_j=ϵq^N-j+1/2-q^-N+j-1/2 q-q^-1v_j,

where ϵ∈{-1,1}, j∈{0,…,N}, v_-1=0, and v_N+1=av_N, where a satisfies <ref>eq.a].

Note that the form of the representation coincide with the classical representations with q^μ=ϵ q^N+1/2, j_1=j_2=r and j_3 determined by the equation Λ_j_3=Λ_0+(q^1/2+q^-1/2)^2A_3. The only differences are following. The equation <ref>eq.Djjk] or the equations <ref>eq.Aj], <ref>eq.Cj] contain an removable singularity for j=N that was caused by expanding the formula for D̃_j by g_j^2. Second difference with the classical representations is the fact that we have D̃_N+1≠ 0 and v_N+1≠ 0, which causes an extra term in matrix representation of I_1 and I_2. In the case when A_1=A_2=0 the matrices are tridiagonal with zero diagonal except the very last entry.

This completes the classification of all representations of (0,0,A_3) for which 2ϵ/(q-q^-1) is not an eigenvector of I_3.

Let q is not a root of unity. Let R be an irreducible representations of _q(0,0,A_3) such that 2ϵ/(q-q^-1) is not an eigenvalue of I_3. Assuming inequality <ref>eq.A3ner] the representation is equivalent to one of the five non-equivalent representations—the classical one given by equations <ref>eq.I1vklas00]–<ref>eq.I3vklas00] or one of the four non-classical ones given by equations <ref>eq.I1vneklas00]–<ref>eq.I3vneklas00]. Assuming equality in the relation <ref>eq.A3ner] for some k, the representation has to be equivalent to one of the four (or three if k=2N+1) non-classical ones given by equations <ref>eq.I1vneklas00]–<ref>eq.I3vneklas00].


sec.ekvivrep]
Dual representations

First of all, we define a representation equivalent to the representation constructed in Section <ref>secc.obrep] multiplying the basis vectors by some scalar. The result will be a bit more symmetric. We define a basis ={x_k}_k=0^N as

    x_j=(1+q^-2μ+2j)∏_k=0^j q^-k+(j_1+j_2+j_3+1)/2A_k-11 (q-q^-1)· v_j.

The equation <ref>eq.I1vobfin] will change to
eq.I1xobfin]

    I_1x_j=- q^ν q-q^-1C_jx_j-1+ q^ν q-q^-1(A_j+C_j-1+q^-2ν)x_j- q^ν q-q^-1A_jx_j+1,

where ν=-μ-1+(j_1+j_2+j_3)/2. The action of I_3 will, of course, not change, so
eq.I3xobfin]

    I_3x_j=-[μ-j]_qx_j.


Now, we will try to construct an equivalent representation that would define a Leonard pair. Consider a representation of algebra _q(A_1,A_2,A_3) defined by equations <ref>eq.I1xobfin], <ref>eq.I3xobfin] and numbers μ, j_1, j_2, and j_3. Our goal is to find an equivalent representation, where I_1 is diagonal and I_3 irreducible tridiagonal. We will make use of the representation of algebra _q(A_3,A_2,A_1) defined by numbers ν=-μ-1+(j_1+j_2+j_3)/2, j_1, j_2, j_3. Substituting into <ref>eq.A3jk]–<ref>eq.A2jk] we can check that it is indeed a representation of algebra with parameters A_3,A_2,A_1. Using isomorphism σ̃=ρ^-1σρ mapping I_1↦ I_3 and I_3↦ I_1 we finally get a representation of _q(A_1,A_2,A_3) we are looking for.

L.dual]
Let μ, j_1, j_2, j_3, ν:=-μ-1+(j_1+j_2+j_3)/2 be complex numbers satisfying the following conditions: q^2μ≠-q^l for every l∈{-1,0,1,…,2N+1}, one of the numbers q^j_1, q^j_2, q^j_3 is equal to q^N+1 and none of the numbers Λ_j_1, Λ_j_2, nor Λ_j_3 is equal to Λ_k for every k∈{1,…,N}. Suppose the same inequalities hold after μ and ν are interchanged. Then the classical irreducible representation of _q(A_1,A_2,A_3), where the parameters A_1, A_2, and A_3 are defined by equations <ref>eq.A3jk]–<ref>eq.A2jk], derived in Section <ref>secc.obrep], is equivalent to a representation defined by the same equations after changing

    I_1↦ I_3,   I_2↦ I_2+I_1I_3-I_3I_1 q^1/2+q^-1/2,   I_3↦ I_1,  μ↦ν,  ν↦μ.

Such a representation will be called dual to the original one.
It is clear that it does not matter if we use the form <ref>eq.I1vobfin] or <ref>eq.I1xobfin], <ref>eq.I3xobfin]. Let us consider the more symmetric form defined above. Then the dual representation has the following form:
eq.I1y]
    I_1y_j=-[ν-j]_qy_j,

eq.I3y]
    I_3y_j=- q^μ q-q^-1D_jy_j-1+ q^μ q-q^-1(B_j+D_j-1+q^-2μ)y_j- q^μ q-q^-1B_jy_j+1,

where
eq.Bj]
    B_j=(1-q^j-j_1+1)(1-q^j-j_2+1)(1-q^j-j_3+1)(1+q^-2ν+j)(1+q^-2ν+2j)(1+q^-2ν+2j+1),

eq.Dj]
    D_j=-q^-2μ(1-q^j)(1+q^-2ν-1+j+j_1)(1+q^-2ν-1+j+j_2)(1+q^-2ν-1+j+j_3)(1+q^-2ν+2j-1)(1+q^-2ν+2j).


As we mentioned, by interchanging ν and μ, we get an irreducible representation of algebra _q(A_3,A_2,A_1). Thus, applying the isomorphism σ̃, we indeed obtain an irreducible representation of _q(A_1,A_2,A_3). If it is classical, then, according to Theorem <ref>V.obecklas], it has to be equivalent to one of the representations we have already found. These are, according to Lemma <ref>L.Rekviv], determined by trace. Hence, we only have to show that the dual representation is classical and that I_3 has the same trace as the original one. By means of direct computation we can check that the trace of I_3 in dual representation is indeed -(q^(N+1)/2-q^-(N+1)/2)/(q^1/2-q^-1/2) [μ-N/2]_q. Now, we show that I_3 has the same eigenvalues in dual representation as in the original one, namely -[μ]_q,…,-[μ-N]_q.

Firstly, we show that -[μ]_q is an eigenvalue of I_3, which means (I_3+[μ]_q)=0. Using the form of the representation <ref>eq.I3y] only, we can show using induction on the dimension r that in the dual representation we have

    (I_3+[μ]_q)=- q^μ q-q^-1∏_k=1^3 (q^1-j_k-1)(q^2-j_k-1)⋯(q^r-j_k-1)(1+q^-2μ+N+1)(1+q^-2μ+N+2)⋯(1+q^-2μ+2N+1).

Using the assumption that q^j_k=q^N+1 for some k we get the zero.

Without loss of generality, we can again assume that the numbers [μ-N+1]_q, [μ-N]_q, …, [μ]_q, [μ+1]_q, …are all mutually different. We can, therefore, repeat the construction of the eigenbasis for the dual representation as well. Let w̃_0 be an eigenvector corresponding to the eigenvalue -[μ]_q. We will apply O_μ+j repeatedly until we get a linearly dependent vector. The last linearly independent vector will be denoted ṽ_0 and the corresponding eigenvalue -[μ̃]_q. Then we define ṽ_j+1:=R_μ̃-jṽ_j. As we mentioned, we cannot have [μ̃-j]_q=[μ̃-j-1]_q, so we have again constructed an eigenbasis {ṽ_0,…,ṽ_N}. The corresponding eigenvalues are -[μ̃]_q,…,-[μ̃-N]_q. Computing their sum, we get the trace of I_3 in dual representation. Comparing with the previous computation we get μ̃=μ.


P.superneklas]
To construct a dual representation, we do not need to assume the existence of the original one. It is sufficient to fulfil assumptions for the existence of classical representation defined by numbers ν, j_1, j_2, and j_3, we do not have to exclude the possibility of q^μ=ϵ q^l/2 for some l∈{0,…,2N+1}. In that case, an eigenbasis for I_3 does not have to exist. Nevertheless, it still holds that the numbers [μ-⌊ l/2⌋]_q,[μ-⌊ l/2⌋+1]_q,… are mutually different, so choosing ṽ_0 an eigenvector corresponding to the eigenvalue -[μ]_q we can define ṽ_j+1=R_μ-jṽ_j until we get 0≠ṽ_⌊ l/2⌋, which is for l even an eigenvector corresponding to the eigenvalue -[μ-l/2]_q=2ϵ/(q-q^-1) and for l odd it is an eigenvector corresponding to -[μ-(l-1)/2]_q=ϵ/(q^1/2-q^-1/2). (The equality v_⌊ l/2⌋=0 would contradict the irreducibility of representation.)

This example shows that for certain parameters of Askey–Wilson algebra there exist representations containing both 2ϵ/(q-q^-1) or ϵ/(q^1/2-q^-1/2 as eigenvalues of I_3.

Correspondence with q-Racah polynomials

Now we are going to show that a representation of Askey–Wilson algebra together with its dual representation defines a Leonard pair corresponding  to q-Racah polynomials. More detailed study of this Leonard pair is available in <cit.>

Firstly, let us recall the explicit formula for q-Racah polynomials that were discovered by Askey and Wilson in <cit.> We use the notation from <cit.> where properties of all orthogonal polynomial series of the Askey–Wilson scheme are summarized.

eq.racdef]

    R_n(μ(x);α,β,γ,δ| q)=43(q^-n,αβ q^n+1,q^-x,γδ q^x+1;α q,βδ q,γ q|q;q)   n=0,1,2,…,N,

where

    μ(x)=q^-x+γδ q^x+1  and

eq.rackondim]
    α q=q^-N  or  βδ q=q^-N  or  γ q=q^-N,  where N∈_0.



Let μ, j_1, j_2, j_3, and ν:=-μ-1+(j_1+j_2+j_3)/2 be numbers satisfying the same assumptions as in the preceding lemma. Define linear operators A,B on V as

    Ax_k=C_kx_k-1-(A_k+C_k-1+q^-2ν)x_k+A_kx_k+1,


    Bx_k=(q^-k-q^-2μ+k)x_k.

Then there exists a basis , where operators A and B have the following form:

    Ay_k=(q^-k-q^-2ν+k)y_k,


    By_k=D_ky_k-1-(B_k+D_k-1+q^-2μ)y_k+B_ky_k+1.

Therefore, the operators A and B form a Leonard pair.
Consider a representation of  in the form <ref>eq.I1xobfin], <ref>eq.I3xobfin]. Then we can write
eq.paircorr]
    A= q^-ν(q-q^-1)I_1,   B= q^-μ(q-q^-1)I_3.

The basis  correspond to dual representation constructed in Lemma <ref>L.dual].


Note that this result agrees with <cit.> Theorem 6.2, which essentially says that irreducible representations of Askey–Wilson algebra define a Leonard pair if both the operators have mutually different eigenvalues.

Now, we can show the correspondence to q-Racah polynomials. Denote
eq.koresp1]
    α=q^-j_1,  β=-q^-2μ-1+j_1,  γ=q^-j_3,  δ=-q^2μ+1-j_1-j_2.

Then the q-Racah polynomials R_n(x), n=0,…,N with parameters α, β, γ, δ are hidden in this Leonard pair in the following way

    _jk=r_jR_j(μ(k)),   r_j=∏_k=0^jA_k-1 C_k,

where  is the transition matrix from basis  to basis  and μ(x)=q^-x+γδ q^x+1.

Indeed, the sequences A_n, B_n, C_n, and D_n can be expressed in terms of α, β, γ, and δ as

    eq.racAn]	A_n   =(1-α q^n+1)(1-αβ q^n+1)(1-βδ q^n+1)(1-γ q^n+1)(1-αβ q^2n+1)(1-αβ q^2n+2),   eq.racBn]	B_n   =(1-α q^n+1)(1-βδ q^n+1)(1-γ q^n+1)(1-γδ q^n+1)(1-γδ q^2n+1)(1-γδ q^2n+2),   eq.racCn]	C_n   =q(1-q^n)(1-β q^n)(γ-αβ q^n)(δ-α q^n)(1-αβ q^2n)(1-αβ q^2n+1),   eq.racDn]	D_n   =q(1-q^x)(1-δ q^n)(β-γ q^n)(α-γδ q^n)(1-γδ q^2n)(1-γδ q^2n+1).

The similarity relations A^= A^ and B^= B^, where A^,A^,B^,B^ denote the matrices of A and B in the bases  and , can be expressed in terms of the q-Racah polynomials as

    A_jR_j+1(μ(k))-(A_j+C_j-1-γδ q)R_j(μ(k))+C_jR_j-1(μ(k))=(q^-k+γδ q^k+1)R_j(μ(k)),


    (q^-j+αβ q^j+1)R_j(μ(k))=D_kR_j(μ(k-1))-(B_k+D_k-1-αβ q)R_j(μ(k))+B_kR_j(μ(k+1)),

which are precisely the three-term recurrence and difference equation for the q-Racah polynomials (cf. <cit.> eqs. (14.2.3) and (14.2.6)). From the three term recurrence we could also easily compute the orthogonality relation.












Now we interpret the assumptions of the Lemma <ref>L.dual] in terms of the orthogonal polynomials sequence. The condition that one of the numbers q^j_i equals to q^N+1 ensuring the finite dimension of the representation can be formulated as
eq.finpol]
    α q=q^-N  or  βδ q=q^-N  or  γ q=q^-N,

which ensures finiteness of the orthogonal polynomials series (<cit.> eq. (14.2.1)). The conditions q^j_i≠ q^k, q^j_i≠ q^2μ-k+1 and q^j_i≠ q^2ν-k+1 for k∈{1,…,N} ensuring irreducibility of the representation and hence irreducibility of the matrices in the Leonard pair can be formulated as follows
eq.irpol1]
    α≠ q^-k,  βδ≠ q^-k,  γ≠ q^-k,

eq.irpol2]
    β≠ q^-k,  α≠δ q^-k,  αβ≠γ q^-k,

eq.irpol3]
    γδ≠α q^-k,  γ≠β q^-k,  δ≠ q^-k.

Although those conditions are usually not mentioned in the literature, they are necessary to obtain orthogonal polynomial sequence with respect to quasi-definite moment functional. If they are not satisfied, one of the coefficients A_n, B_n, C_n, or D_n may be zero for certain n. See also the orthogonality relation (14.2.2) in <cit.>

The last assumption that q^2μ≠ -q^l and q^2ν≠ -q^l for all l∈{-1,0,1,…,2N+1} was made a priori to ensure that the representation is classical and therefore diagonalizable. In terms of the parameters α, β,γ,δ it means
eq.classpol]
    αβ q≠ q^-l  and  γδ q≠ q^-l.

Looking at the formulas <ref>eq.racAn]–<ref>eq.racDn] it seems that those conditions are necessary for the Leonard pair to be well defined since otherwise there may be zero in one of the denominators. Nevertheless if αβ q=q^-l or γδ q=q^-l for l∈{-1,0,2N,2N+1} the singularity is removable. However, for l∈{1,…,2N-1} the condition is indeed necessary, which is also usually not explicitly stated in the literature. Note also that, for example, if we had αβ q=q^-l for l∈{1,…,2N-1}, the n-th polynomial R_n would not be of degree n and the N-tuple would not be linearly independent.

Non-classical representations

From the formulas for the Askey–Wilson polynomials, we can now go backwards and derive the missing non-classical representations. Those shell have the same form as the classical representations except for some changes (cf. Section <ref>secc.00]).

First of all I_3 has again the spectrum [μ]_q, [μ-1]_q, …, [μ-N]_q. For q^2μ=-q^l, l∈{-1,0,2N,2N+1} the eigenvalues are pairwise distinct. In the end, we can show that the representation with l=-1 is equivalent to the representation with l=2N+1 and the representation l=0 is equivalent with representation l=2N (as in Lemma <ref>L.Rekviv]). Thus, we will work now only with the cases l∈{2N,2N+1}.

Take the non-classical case l=2N+1 and define I_1 by formula <ref>eq.I1xobfin] for j_1, j_2, j_3 satisfying the standard assumptions as in Theorem <ref>V.obecklas] and q^mu=± q^N+1/2. Since C_0=0, we do not have to determine the value of x_-1. Nevertheless, we have A_N≠ 0, so we have to determine x_r. By analogy with Section <ref>secc.00], we can guess that x_N+1=ax_N. So, we can substitute into <ref>eq.I1xobfin]

    q^-ν(q-q^-1)I_1x_N=C_Nx_N-1-((1-a)A_N+C_N-1+q^-2ν)x_N.

Such form would lead to three-term recurrence

    ((1-a)A_N+C_N-1-γδ q)R_N(μ(k))+C_NR_N-1(μ(k))=(q^-k+γδ q^k+1)R_N(μ(k))


We can suppose that this should not contradict the standard form of three-term recurrence. Notice that for αβ q=q^-2N-1 we have R_N=R_N+1, so we can rewrite the three-term recurrence as

    -(C_N-1-γδ q)R_N(μ(k))+C_NR_N-1(μ(k))=(q^-k+γδ q^k+1)R_N(μ(k)).


From this, we can conclude that x_N+1=x_N. Finally, we can check that the formulas for classical representation also define a non-classical representation for q^2μ=-q^2N+1 if we define x_N+1=x_N.

In the case l=2N we have αβ q=q^-2N and by similar reasoning we can conclude that x_N+1=x_N-1.

Correspondence with Huang's classification

As we mentioned in the introduction, a complete classification of so-called universal Askey–Wilson algebra appeared in <cit.> The generating elements are represented by two bidiagonal and one tridiagonal matrices. From those explicit formulas we have the following.

For any irreducible representation of the Askey–Wilson algebra there exists μ∈ such that spectrum of I_3 is {-[μ-j]}_j=0^N. On the other hand, for any μ∈ there exists such a representation for suitable parameters.


The paper also gives the criterion for diagonalizability of the generating elements.

(<cit.> Lemma 4.6) Let R be an irreducible representation of . Then all eigenspaces of I_3 are one-dimensional

Corollary(<cit.> Lemma 5.1). Let R be an irreducible representation of  and denote μ such that {-[μ-j]}_j=0^N is the spectrum of I_3. Then the following are equivalent.
n
* I_3 is diagonalizable,
* the numbers [μ], [μ-1], …, [μ-N] are pairwise distinct,
* q^2μ≠-q^l, l∈{1,…,2N-1}.


From these propostions we can see that the assuption that the representation is classical (i.e. 2ϵ/(q-q^-1) and ϵ/(q^1/2-q^-1/2) are not eigenvalues of I_3) is sufficient to ensure diagonalizability of I_3, but not necessary.

Therefore, our classification contains all diagonalizable representations of the Askey–Wilson algebra except the “border cases”, when q^2μ∈{q^-1,q^0,q^2N,q^2N+1}, which we discussed in the previous section.











Conclusion

We classified representations of Askey–Wilson algebra satisfying certain conditions allowing us to use the shift operators to construct an eigenbasis of I_3. Representations satisfying similar condition for another generating element can be obtained by applying suitable isomorphism of the Askey–Wilson algebra as we indicated in Section <ref>sec.ekvivrep]. Those representations are very important since they define the Leonard pair connected to the q-Racah polynomials. In such way, q-Racah polynomials (and other types of orthogonal polynomials in the Askey scheme) can be then obtained “at no cost” and this is the most valuable side-effect of solving the classification problem of Askey–Wilson algebra.



Acknowledgements

This work was supported by the Grant Agency of the Czech Technical University in Prague, grant numbers SGS15/215/OHK4/3T/14 and SGS16/239/OHK4/3T/14.

References


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