The $q$-Onsager Algebra and the Universal Askey-Wilson Algebra

Recently Pascal Baseilhac and Stefan Kolb obtained a PBW basis for the $q$-Onsager algebra $\mathcal O_q$. They defined the PBW basis elements recursively, and it is obscure how to express them in closed form. To mitigate the difficulty, we bring in the universal Askey-Wilson algebra $\Delta_q$. There is a natural algebra homomorphism $\natural \colon \mathcal O_q \to \Delta_q$. We apply $\natural $ to the above PBW basis, and express the images in closed form. Our results make heavy use of the Chebyshev polynomials of the second kind.


Introduction
In the 1944 paper [25] Lars Onsager obtained the free energy of the two-dimensional Ising model in a zero magnetic field. In that paper an infinite-dimensional Lie algebra was introduced; this algebra is now called the Onsager algebra and denoted by O. Onsager defined his algebra by giving a linear basis and the action of the Lie bracket on the basis. In [26] Perk gave a presentation of O by generators and relations. This presentation involves two generators and two relations, called the Dolan/Grady relations [17]. This presentation is discussed in [30,Remark 9.1]. Via this presentation, the universal enveloping algebra of O admits a q-deformation O q called the q-Onsager algebra [4,29]. The algebra O q is associative and infinite-dimensional. It is defined by two generators and two relations called the q-Dolan/Grady relations; these are given in (2.2), (2.3) below. The q-Dolan/Grady relations first appeared in algebraic combinatorics, in the study of Q-polynomial distance-regular graphs [27,Lemma 5.4]. Shortly thereafter they appeared in physics, in the study of statistical mechanical models [4,Section 2]. Up to the present, the representation theory of O q remains an active area of research in mathematics [19,21,22,28,29,30,31,32,33] and physics [3,4,5,6,7,8,9,10,11,12,14,15]. This theory involves a linear algebraic object called a tridiagonal pair [20]. A finite-dimensional irreducible O q -module is essentially the same thing as a tridiagonal pair of q-Racah type [29,Theorem 3.10]. These tridiagonal pairs are classified up to isomorphism in [21,Theorem 3.3]. In [22,Theorem 2.1], Ito and the present author gave a linear basis for O q , called the zigzag basis. More information about this basis can be found in [32,Note 4.7]. In [7], Baseilhac and Belliard conjectured another linear basis for O q ; this one is motivated by how O q is related to the reflection equation algebra [11,14]. In [13], Baseilhac and Kolb introduced two automorphisms T 0 , T 1 of O q that are roughly analogous to the Lusztig automorphisms of U q ( sl 2 ). They used T 0 , T 1 and a method of Damiani [16] to obtain a Poincaré-Birkhoff-Witt (or PBW) basis for O q [13,Theorem 4.3]. In our view this PBW basis is important and worthy of further This paper is a contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications (OPSFA14). The full collection is available at https://www.emis.de/journals/SIGMA/OPSFA2017.html study. In the present paper we study the following aspect. In [13, Section 3.1] the PBW basis is defined recursively, and it is obscure how to express it in closed form. In order to mitigate the difficulty, we bring in a related algebra which we now describe. In [34] Zhedanov introduced the Askey-Wilson algebra AW(3) and used it to describe the Askey-Wilson polynomials. In [31] the present author introduced a central extension of AW(3), called the universal Askey-Wilson algebra ∆ q . In [18], Hau-Wen Huang classified up to isomorphism the finite-dimensional irreducible ∆ q -modules for q not a root of unity. A linear basis for ∆ q is given in [31,Theorem 7.5]. There is a natural algebra homomorphism : O q → ∆ q [31,Definition 10.4]; this is described below (2.23) in the present paper. We use to describe the PBW basis for O q in the following way. We apply to the PBW basis vectors and consider their images in ∆ q . We express these images explicitly in the linear basis for ∆ q mentioned above. Our main results are Theorems 5.5, 5.6. These results make heavy use of the Chebyshev polynomials of the second kind [23,24].

Preliminaries
We now begin our formal argument. Recall the natural numbers N = {0, 1, 2, . . .} and integers Z = {0, ±1, ±2, . . .}. Let F denote an algebraically closed field with characteristic zero. All the algebras discussed in this paper are over F; those without the Lie prefix are associative and have a multiplicative identity. Fix a nonzero q ∈ F that is not a root of 1. Recall the notation We will be discussing the q-Onsager algebra O q and the universal Askey-Wilson algebra ∆ q . We now recall these algebras. The algebra O q (see [4,Section 2], [29,Definition 3.9]) is defined by generators A, B and relations The relations (2.2), (2.3) are called the q-Dolan/Grady relations. In [13], Baseilhac and Kolb introduced the automorphisms T 0 , T 1 of O q . These automorphisms satisfy The inverse automorphisms satisfy In [13], Baseilhac and Kolb used T 0 and T 1 to define some elements in O q , denoted . . . . . . and for n ≥ 1, (2.10) Next we recall the universal Askey-Wilson algebra ∆ q [31, Definition 1.2]. This algebra is defined by generators and relations. The generators are A, B, C. The relations assert that each of the following is central in ∆ q : For the above three central elements, multiply each by q + q −1 to get α, β, γ. Thus (2.13) Each of α, β, γ is central in ∆ q . By [31,Corollary 8.3] the center Z(∆ q ) is generated by α, β, γ, Ω where (2.14) The element Ω is called the Casimir element. By [31, Theorem 8.2] the following is a linear basis for the F-vector space Z(∆ q ): Ω α r β s γ t , , r, s, t ≥ 0.
We mention two bases for ∆ q . By [31, Theorem 4.1], the following is a linear basis for the F-vector space ∆ q : By [31,Theorem 7.5], the following is a linear basis for the F-vector space ∆ q : For convenience we will work with the basis (2.16).
Shortly we will discuss how ∆ q is related to O q . To aid in this discussion we recall from [31, Section 2] a second presentation of ∆ q . By (2.11)-(2.13) the algebra ∆ q is generated by A, B, γ. Moreover By [31, Theorem 2.2] the algebra ∆ q has a presentation by generators A, B, γ and relations In order to clarify the nature of T 0 , T 1 , we now introduce some automorphisms t 0 , t 1 of ∆ q such that t 0 = T 0 and t 1 = T 1 . To this end, we recall from [31, Section 3] how the modular group PSL 2 (Z) acts on ∆ q as a group of automorphisms. By [1] the group PSL 2 (Z) has a presentation by generators ρ, σ and relations ρ 3 = 1, σ 2 = 1. Earlier in this section we gave two presentations of ∆ q . Using these presentations we find that PSL 2 (Z) acts on ∆ q as a group of automorphisms in the following way: This action is faithful by [31,Theorem 3.13]. From the table (2.24) we see that the PSL 2 (Z)generators ρ, σ each permute α, β, γ. This gives a group homomorphism from PSL 2 (Z) onto the symmetric group S 3 . Let P denote the kernel of the homomorphism. Thus P is a normal subgroup of PSL 2 (Z), and the quotient group We remark that in the literature the groups PSL 2 (Z) and P are often denoted by Γ and Γ(2), respectively; see for example [1,2]. Define The actions (2.26)-(2.29) match (2.4)-(2.7). Consequently the following diagrams commute: Using the commuting diagrams (2.30) one finds that for π ∈ P the following diagram commutes: (2.31) We now prove that ε is an isomorphism. By construction ε is surjective. We show that ε is injective. Given an element r in the kernel of ε, we show that r is the identity in P. To this end, we show that r fixes the generators A, B, γ of ∆ q . The map ε(r) is the identity in G, so ε(r) fixes the elements A, B of O q . By the commuting diagram (2.31) the map r fixes the elements A, B of ∆ q . Also r fixes γ since r ∈ P and everything in P fixes γ. We have shown that r fixes the generators A, B, γ of ∆ q so r is the identity in P. Consequently ε is injective and hence an isomorphism.
It is mentioned in [13, Section 2.3] that one expects G to be freely generated by T ±1 0 , T ±1 1 . This is now easily proven as follows. The group P is freely generated by t ±1 0 , t ±1 1 . Applying the isomorphism ε : P → G we find that G is freely generated by T ±1 0 , T ±1 1 . Next we consider how the map : O q → ∆ q acts on the elements (2.8). For these elements we retain the same notation for their images under . Our goal is to obtain these images in closed form, in terms of the basis (2.16). In order to obtain these images, it is convenient to bring in the Chebyshev polynomials of the second kind. These polynomials are reviewed in the next section.

The Chebyshev polynomials
In this section we review the Chebyshev polynomials of the second kind; see [23,24] for further details. Let x denote an indeterminate. Let F[x] denote the F-algebra consisting of the polynomials in x that have all coefficients in F. Definition 3.1 (see [24, p. 4]). For n ∈ N define U n ∈ F[x] by The polynomial U n is monic and degree n. We call U n the nth Chebyshev polynomial of the second kind. For notational convenience define U n = 0 for all integers n < 0.
Note 3.2. The above polynomials U n are normalized to be monic. This normalization differs from the one in [23, Section 9.8.2]. To go from our normalization to the one in [23, Section 9.8.2], replace x by 2x.
In the table below we display U n for 0 ≤ n ≤ 9.
By [24, pp. 332-333], Next we express the polynomials U n in a more closed form.
where we recall x = z + z −1 .
In this paper, on several occasions we will consider generating functions in an indeterminate t. These generating functions involve a formal power series; issues of convergence are not considered. The following generating function will be useful. (3.1) Proof . Using Definition 3.1 one finds that the product n∈N t n U n (x) 1 − tx + t 2 is equal to 1.
The following variations on Lemma 3.4 will be used repeatedly.
Lemma 3.5. For an indeterminate t, Lemma 3.6. For an indeterminate t,

Some identities
In this section we give some identities for later use.

Lemma 4.3. For an indeterminate t,
Proof . These are readily checked.

The main results
In this section we express the images (2.8) in the basis (2.16). In what follows, the notation [u, v] means uv − vu. We will use a recursion found in [13]; we give a short proof for the sake of completeness.
The following is our first main result.
Theorem 5.5. For n ≥ 0 the following hold in ∆ q : Proof . By a routine induction on n, using Lemmas 5.3, 5.4.
The following is our second main result.
Theorem 5.6. In the algebra ∆ q , for n ≥ 1 the element B nδ is equal to (−1) n 1 − q −2 times a weighted sum with the following terms and coefficients: Proof . We have some preliminary comments. Using (2.12), (2.13), By [31, Lemma 6.1], We are done with the preliminary comments. We now define some generating functions in an indeterminate t: (5.14) By (2.9), By (5.10), (5.11), We next consider what the second equation in Theorem 5.5 implies about Φ(t). Using Lemma 3.4, Using Lemma 3.5, We have Similarly, By these comments and the second equation in Theorem 5.5, By (5.16) and (5.17), In (5.15), we multiply each side on the left by q −1 t+qt −1 +C and on the right by qt+q −1 t −1 +C. We evaluate the result using (5.17), (5.18) to obtain Evaluating the above equation using the preliminary comments, we find that Consequently Ψ(t) is equal to 1 − q −2 times We now compare the {F i } 5 i=1 with the coefficients shown in the table of the theorem statement.
and also Note that (5.20) plus twice (5.21) is equal to F 4 (t, x).
Recall the center Z(∆ q ).
Corollary 5.7. For n ≥ 1 the element B nδ is contained in the subalgebra of ∆ q generated by C and Z(∆ q ).
We finish the paper with some comments. Here is another version of Theorem 5.5.
Proof . Similar to the proof of Theorem 5.5.
The following result might be of independent interest. Proposition 5.9. For n ≥ 1 the following holds in ∆ q : and also Proof . We use induction on n. For n = 1 the equations in the proposition statement are reformulations of (2.11), (2.12). For n ≥ 2 we proceed as follows. To obtain the first (resp. second) equation in the proposition statement, multiply each side of (2.12) (resp. (2.11)) on the left by U n−1 (C), and evaluate the result using CU n−1 (C) = U n (C) + U n−2 (C) along with induction and Lemmas 4.1, 4.2.
In the algebra O q the elements {B nδ } ∞ n=1 are defined using the formula (2.9). This formula is not symmetric in α 0 , α 1 . As shown in [13], there is another formula for {B nδ } ∞ n=1 that interchanges the roles of α 0 , α 1 . According to [13, Section 5.2] the following holds in O q for n ≥ 1: We now sketch a proof of Theorem 5.6 that uses (5.25) instead of (2.9). Following (5.14), for the algebra ∆ q we definẽ Φ(t) =