The Universal Askey-Wilson Algebra and DAHA of Type $(C_1^{\vee},C_1)$

Let $\mathbb F$ denote a field, and fix a nonzero $q\in\mathbb F$ such that $q^4\not=1$. The universal Askey-Wilson algebra $\Delta_q$ is the associative $\mathbb F$-algebra defined by generators and relations in the following way. The generators are $A$, $B$, $C$. The relations assert that each of $A+\frac{qBC-q^{-1}CB}{q^2-q^{-2}}$, $B+\frac{qCA-q^{-1}AC}{q^2-q^{-2}}$, $C+\frac{qAB-q^{-1}BA}{q^2-q^{-2}}$ is central in $\Delta_q$. The universal DAHA $\hat H_q$ of type $(C_1^\vee,C_1)$ is the associative $\mathbb F$-algebra defined by generators $\lbrace t^{\pm1}_i\rbrace_{i=0}^3$ and relations (i) $t_i t^{-1}_i=t^{-1}_i t_i=1$; (ii) $t_i+t^{-1}_i$ is central; (iii) $t_0t_1t_2t_3=q^{-1}$. We display an injection of $\mathbb F$-algebras $\psi:\Delta_q\to\hat H_q$ that sends $A\mapsto t_1t_0+(t_1t_0)^{-1}$, $B\mapsto t_3t_0+(t_3t_0)^{-1}$, $C\mapsto t_2t_0+(t_2t_0)^{-1}$. For the map $\psi$ we compute the image of the three central elements mentioned above. The algebra $\Delta_q$ has another central element of interest, called the Casimir element $\Omega$. We compute the image of $\Omega$ under $\psi$. We describe how the Artin braid group $B_3$ acts on $\Delta_q$ and $\hat H_q$ as a group of automorphisms. We show that $\psi$ commutes with these $B_3$ actions. Some related results are obtained.


Introduction
The Askey-Wilson polynomials were introduced in [2] and soon became a major topic in special functions [6,9].This topic became linked to representation theory through the work of A. Zhedanov [19].In that article Zhedanov introduced the Askey-Wilson algebra AW(3), and showed that its "ladder" representations give the Askey-Wilson polynomials.The algebra AW(3) is noncommutative and infinite-dimensional.It is defined by generators and relations.The relations involve a scalar parameter q and a handful of extra scalar parameters.The number of extra parameters ranges from 3 to 7 depending on which normalization is used [12,Line (6.1)], [18,Section 4.3], [17,Theorem 1.5], [19, lines (1.1a)-(1.1c)].In [15] we introduced a central extension of AW (3), denoted ∆ q and called the universal Askey-Wilson algebra.Up to normalization ∆ q is obtained from AW (3) by interpreting the extra parameters as central elements in the algebra.By construction ∆ q has no scalar parameters besides q, and there is a surjective algebra homomorphism ∆ q → AW(3).One advantage of ∆ q over AW(3) is that ∆ q has a larger automorphism group.Our definition of ∆ q was inspired by [7, Section 3], which in turn was motivated by [5].
In [15] we began a comprehensive investigation of ∆ q .In that paper we focused on its ring-theoretic aspects, and in a followup paper [16] we related ∆ q to the quantum algebra U q (sl 2 ).In the present paper we relate ∆ q to the universal DAHA of type (C ∨ 1 , C 1 ) [7].Broadly speaking, the results in the paper amount to a universal analog of the results of Koornwinder [10,11] which relate AW (3) to the DAHA of type (C ∨ 1 , C 1 ).We view [10,11] as groundbreaking and the main inspiration for the present paper.
We will describe our results after we summarize the contents of [15], [16].
Our conventions for the paper are as follows.An algebra is meant to be associative and have a 1.A subalgebra has the same 1 as the parent algebra.Fix a field F. All unadorned tensor products are meant to be over F. We fix a nonzero q ∈ F such that q 4 = 1.Recall the natural numbers N = {0, 1, 2 . ..} and integers Z = {0, ±1, ±2 . ..}.
The universal Askey-Wilson algebra ∆ q is the F-algebra defined by generators and relations in the following way.The generators are A, B, C. The relations assert that each of is central in ∆ q .For the central elements (1) multiply each by q + q −1 to get α, β, γ.
In [15] we obtained the following results about ∆ q .We gave an alternate presentation for ∆ q by generators and relations; the generators are A, B, γ.We gave a faithful action of the modular group PSL 2 (Z) on ∆ q as a group of automorphisms; one generator sends (A, B, C) → (B, C, A) and another generator sends (A, B, γ) → (B, A, γ).We showed that {A i B j C k α r β s γ t |i, j, k, r, s, t ∈ N} is a basis for the F-vector space ∆ q .We showed that the center Z(∆ q ) contains a Casimir element Ω = q −1 ACB + q −2 A 2 + q −2 B 2 + q 2 C 2 − q −1 Aα − q −1 Bβ − qCγ.
Under the assumption that q is not a root of unity, we showed that Z(∆ q ) is generated by Ω, α, β, γ and that Z(∆ q ) is isomorphic to a polynomial algebra in four variables.
In [16] we relate ∆ q to the quantum algebra U q (sl 2 ).To describe this relationship we use the equitable presentation for U q (sl 2 ) which was introduced in [8].This equitable presentation has generators x, y ±1 , z and relations yy −1 = y −1 y = 1, qxy − q −1 yx q − q −1 = 1, qyz − q −1 zy q − q −1 = 1, Let a, b, c denote mutually commuting indeterminates.Let F[a ±1 , b ±1 , c ±1 ] denote the Falgebra of Laurent polynomials in a, b, c that have all coefficients in F. In [16, Theorems 2.16, 2.18] we displayed an injection of F-algebras ♮ : ∆ q → U q (sl 2 ) ⊗ F[a ±1 , b ±1 , c ±1 ] that sends The map ♮ sends where Λ = qx + q −1 y + qz − qxyz is the normalized Casimir element of U q (sl 2 ) [16, Section 2].In [16,Theorem 2.17] we showed that ♮ sends Ω to We now summarize the present paper.We first show that the following is a basis for the F-vector space ∆ q : This basis plays a role in our main topic, which is about how ∆ q is related to the universal DAHA Ĥq of type (C ∨ 1 , C 1 ).The algebra Ĥq is a variation on an algebra Ĥ introduced in [7].By definition Ĥq is the F-algebra with generators {t ±1 i } 3 i=0 and relations (i) t i t −1 i = t −1 i t i = 1; (ii) t i + t −1 i is central; (iii) t 0 t 1 t 2 t 3 = q −1 .We display an injection of F-algebras ψ : ∆ q → Ĥq that sends We show that ψ sends Ω to We remark that for the above results some parts are easier to prove than others.It is relatively easy to show that ψ exists as an algebra homomorphism.Indeed this existence essentially follows from [7,Theorem 5.2], although in our formal argument we take another approach which quickly yields the result from first principles.We found it relatively hard to show that ψ is injective; indeed this argument takes up the majority of the paper.To establish injectivity we display a basis for Ĥq , and use it to show that ψ sends the basis (2) to a linearly independent set.Adapting [5,Theorem 2.6], [7,Lemma 4.2] we show how the Artin braid group B 3 acts on Ĥq as a group of automorphisms.The group B 3 is a homomorphic preimage of PSL 2 (Z), and we mentioned earlier that PSL 2 (Z) acts on ∆ q as a group of automorphisms.Pulling back this PSL 2 (Z) action we get a B 3 action on ∆ q as a group of automorphisms.We show that ψ commutes with the B 3 actions for ∆ q and Ĥq .Now consider the image of ∆ q under ψ.We show that the subalgebra {h ∈ Ĥq |t 0 h = ht 0 } is generated by this image together with t 0 and {t i + t −1 i } 3 i=1 .For this subalgebra we give a presentation by generators and relations.This presentation amounts to a q-analog of [13, Theorem 2.1] and a universal analog of [10,Corollary 6.3], [11,Theorem 5.1].Under the assumption that q is not a root of unity, we show that Z( Ĥq ) is generated by {t i + t −1 i } 3 i=0 and that Z( Ĥq ) is isomorphic to a polynomial algebra in four variables.
The paper is organized as follows.In Section 2, after reviewing ∆ q we obtain a basis for this algebra that will be useful.In Section 3 we define Ĥq and discuss its symmetries.In Section 4 we state five theorems which describe an injection ψ : ∆ q → Ĥq ; these are Theorems 4.1-4.5.In Section 5 we establish some identities in Ĥq that will be used repeatedly.In Section 6 we prove Theorems 4.1, 4.2, 4.3.In Section 7 we display a basis for Ĥq , along with some reduction rules that show how to write any given element of Ĥq in the basis.Sections 8, 9 are devoted to proving Theorem 4.4.Sections 10-12 are devoted to proving Theorem 4.5.In Section 13 we consider the image of ∆ q under the map ψ.We show that the subalgebra {h ∈ Ĥq |t 0 h = ht 0 } is generated by this image together with t 0 and {t i + t −1 i } 3 i=1 .In Section 14 we give a presentation for this subalgebra by generators and relations.In Section 15 we describe Z( Ĥq ).

The universal Askey-Wilson algebra
We now begin our formal argument.In this section we discuss the universal Askey-Wilson algebra.After reviewing its basic features we establish a basis for the algebra that will be useful later in the paper.
is central in ∆ q .The algebra ∆ q is called the universal Askey-Wilson algebra.
Definition 2.2 [15,Definition 1.3].For the three central elements in (3), multiply each by q + q −1 to get α, β, γ.Thus Note that each of α, β, γ is central in ∆ q .Note also that A, B, γ is a generating set for ∆ q .
We now discuss some automorphisms of ∆ q .Recall that the modular group PSL 2 (Z) has a presentation by generators p, s and relations p 3 = 1, s 2 = 1.See for example [1].
The group PSL 2 (Z) acts on ∆ q as a group of automorphisms in the following way: The group PSL 2 (Z) has a central extension called the Artin braid group B 3 .The group B 3 is defined as follows.
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 ∆ q , to get a B 3 action on ∆ q as a group of automorphisms.This action is described as follows.
Lemma 2.5 The group B 3 acts on ∆ q as a group of automorphisms such that τ (h) = h for all h ∈ ∆ q and ρ, σ do the following: In Definition 2.2 we defined the central elements α, β, γ of ∆ q .There is another central element of interest, called the Casimir element Ω.This element is defined as follows.
Lemma 2.7 [15,Theorems 6.2,8.2].The Casimir element Ω is contained in the center Z(∆ q ).Moreover {Ω i α r β s γ t |i, r, s, t ∈ N} is a basis for the F-vector space Z(∆ q ), provided that q is not a root of unity.
Given an F-algebra A, by an antiautomorphism of A we mean an F-linear bijection ζ : Lemma 2.9 There exists an antiautomorphism † of ∆ q that sends Moreover † 2 = 1.
Proof: This follows from [15,Lemma 6.1].✷ We are going to display a basis for the F-vector space ∆ q .Two such bases can be found in [15, Theorems 4.1, 7.5], but these are not suited for our present purpose.To obtain a suitable basis we work with the following presentation of ∆ q .
Proposition 2.13 The F-algebra ∆ q is presented by generators and relations in the following way.The generators are A, B, C, Ω, α, β, γ.The relations assert that each of Ω, α, β, γ is central and Proof: Referring to the above four equations, the first three are reformulations of ( 4)-( 6) and the fourth is a reformulation of (7).✷ Definition 2.14 The generators A, B, C, Ω, α, β, γ of ∆ q are called balanced.
Note 2.15 Referring to the presentation of ∆ q from Proposition 2.13, consider the relations which assert that Ω, α, β, γ are central.These relations can be expressed as Definition 2.16 By a reduction rule for ∆ q we mean an equation which appears in Proposition 2.13 or Note 2.15.A reduction rule from Proposition 2.13 is said to be of the first kind.A reduction rule from Note 2.15 is said to be of the second kind.
Definition 2.17 For an integer n ≥ 0, by a word of length n in ∆ q we mean a product g 1 g 2 • • • g n such that g i is a balanced generator of ∆ q for 1 ≤ i ≤ n.We interpret the word of length 0 as the multiplicative identity in ∆ q .A word is called forbidden whenever it is the left-hand side of a reduction rule.Every forbidden word has length two.A forbidden word is said to be of the first kind (resp.second kind) whenever the corresponding reduction rule is of the first (resp.second) kind.
Definition 2.18 Let w denote a forbidden word in ∆ q , and consider the corresponding reduction rule.By a descendent of w we mean a word that appears on the right-hand side of that reduction rule.Theorem 2.20 The following is a basis for the F-vector space ∆ q : Proof: We invoke Bergman's Diamond Lemma [3,Theorem 1.2].Let g 1 g 2 • • • g n denote a word in ∆ q .This word is called reducible whenever there exists an integer i (2 ≤ i ≤ n) such that g i−1 g i is forbidden.The word is called irreducible whenever it is not reducible.The list (8) consists of the irreducible words in ∆ q .Let w = g 1 g 2 • • • g n denote a word in ∆ q .By an inversion in w we mean an ordered pair of integers (i, j) such that 1 ≤ i < j ≤ n and the word g i g j is forbidden.The inversion (i, j) is of the first kind (resp.second kind) whenever the forbidden word g i g j is of the first kind (resp.second kind).Let W denote the set of all words in ∆ q .We define a partial order < on W as follows.Pick any words w, w ′ in W and write w = g 1 g 2 • • • g n .We say that w dominates w ′ whenever there exists an integer i (2 ≤ i ≤ n) such that (i − 1, i) is an inversion for w, and w ′ is obtained from w by replacing g i−1 g i by one of its descendents.In this case either (i) w has more inversions of the first kind than w ′ , or (ii) w and w ′ have the same number of inversions of the first kind, but w has more inversions of the second kind than w ′ .By these comments the transitive closure of the domination relation on W is a partial order on W which we denote by <.By construction < is a semigroup partial order [3, p. 181] and satisfies the descending chain condition [3, p. 179].We now relate the partial order < to our reduction rules.Let w = g 1 g 2 • • • g n denote a reducible word in ∆ q .Then there exists an integer i (2 ≤ i ≤ n) such that g i−1 g i is forbidden.There exists a reduction rule with g i−1 g i on the left-hand side; in w we eliminate g i−1 g i using this reduction rule and thereby express w as a linear combination of words, each less than w with respect to <. Therefore the reduction rules are compatible with < in the sense of Bergman [3,p. 181].In order to employ the Diamond Lemma, we must show that the ambiguities are resolvable in the sense of Bergman [3,p. 181].There are potentially two kinds of ambiguities; inclusion ambiguities and overlap ambiguities [3, p. 181].For the present example there are no inclusion ambiguities.The nontrivial overlap ambiguities are Take for example BCA.The words BC and CA are forbidden.Therefore BCA can be reduced in two ways; we could evaluate BC first or we could evaluate CA first.Either way, after a 4-step reduction we get the same resolution, which is q −3 (q 2 − q −2 )Ω + q −6 ACB − q −3 (q 4 − q −4 )A 2 − q −3 (q 4 − q −4 )B 2 + q −3 (q 3 − q −3 )Aα + q −3 (q 3 − q −3 )Bβ + q −3 (q − q −1 )Cγ.
Therefore the ambiguity BCA is resolvable.The ambiguities BC 2 and C 2 A are similarly shown to be resolvable.The resolution of BC 2 is We conclude that every ambiguity is resolvable, so by the Diamond Lemma [3, Theorem 1.2] the irreducible words form a basis for ∆ q .The result follows.✷ 3 The universal DAHA Ĥq type The double affine Hecke algebra (DAHA) for a reduced root system was defined by Cherednik [4], and the definition was extended to include nonreduced root systems by Sahi [14].
The most general DAHA of rank 1 is associated with the root system (C ∨ 1 , C 1 ).In [7] we introduced a central extension of this algebra called the universal DAHA of type (C ∨ 1 , C 1 ).In the present paper we will work with a variation on this algebra.
For notational convenience define a four element set The following definition is a variation on [7, Definition 3.1].Definition 3.1 Let Ĥq denote the F-algebra defined by generators {t ±1 i } i∈I and relations We call Ĥq the universal DAHA of type (C ∨ 1 , C 1 ).
Remark 3.2 In [7, Definition 3.1] we defined an F-algebra Ĥ by generators {t ±1 i } i∈I and relations (i) The following two lemmas are immediate from Definition 3.1.

Lemma 3.3
In the algebra Ĥq the scalar q −1 is equal to each of the following: Lemma 3.4 There exists an automorphism of Ĥq that sends Recall the braid group B 3 from Definition 2.4.

Lemma 3.5
The group B 3 acts on Ĥq as a group of automorphisms such that τ (h) = t −1 0 ht 0 for all h ∈ Ĥq and ρ, σ do the following: Proof: This is routinely checked, or see [7,Lemma 4.2].✷ Lemma 3.6 The B 3 action on Ĥq does the following to the central elements (10).The generator τ fixes every central element.The generators ρ, σ satisfy the table below.
Proof: Use Definition 3.1.✷ Lemma 3.8 There exists a unique isomorphism of F-algebras ξ : Ĥq → Ĥq −1 that sends 4 How ∆ q is related to Ĥq In this section we state five theorems concerning how ∆ q is related to Ĥq .The proofs of these theorems will take up most of the rest of the paper.
Theorem 4.1 There exists a unique F-algebra homomorphism ψ : ∆ q → Ĥq that sends The homomorphism ψ sends ). Theorem 4.2 For all g ∈ B 3 the following diagram commutes: The following diagrams commute: Theorem 4.4 Under the homomorphism ψ from Theorem 4.1 the image of Ω is Theorem 4.5 The homomorphism ψ from Theorem 4.1 is injective.

Preliminaries concerning Ĥq
In this section we establish some basic facts about Ĥq .These facts will used repeatedly for the rest of the paper.
Definition 5.1 For the algebra Ĥq define Note that each T i is central in Ĥq .
In Definition 3.1 we gave a presentation for Ĥq involving the generators In terms of the generators {t i } i∈I , {T i } i∈I the algebra Ĥq looks as follows.
Lemma 5.2 The F-algebra Ĥq has a presentation by generators {t i } i∈I , {T i } i∈I and relations Definition 5.3 Let X, Y denote the following elements of Ĥq : Note that each of X, Y is invertible.
Proof: The relations (15) are routinely checked using Definition 3.1 and (14).✷ In terms of the generators X ±1 , Y ±1 , t ±1 0 the {T i } i∈I look as follows.
Lemma 5.5 For the algebra Ĥq the following (i)-(iv) hold.
(iv) T 3 is equal to each of Proof: (i) Clear.
(ii) Using the equation on the left in (15), Also (iii) Using the middle equation in (15), Also (iv) Similar to the proof of (ii) above.✷ In Section 3 we discussed some automorphisms and antiautomorphisms of Ĥq .We now consider how these maps act on X, Y .The following four lemmas are routinely checked.
Lemma 5.9 Recall the isomorphism ξ : Ĥq → Ĥq −1 from Lemma 3.8.This map sends We now give some relations that show how t 0 commutes past the X ±1 , Y ±1 .

Lemma 5.11
The following relations hold in Ĥq : Proof: Concerning line (20), the equation on the left comes from The equation on the right comes from To obtain (21)-( 23), repeatedly apply the automorphism from Lemma 3.4 to everything in (20), and use Lemma 5.6.✷ Definition 5.12 Let {C i } i∈I denote the following elements in Ĥq : Lemma 5. 13 The automorphism from Lemma 3.4 sends Proof: Use Lemma 5.6 and Definition 5.12.✷ Proposition 5. 14 The following relations hold in Ĥq : Proof: To verify (24), use (23) together with To verify (25)-( 27), repeatedly apply the automorphism from Lemma 3.4 to everything in (24), and use Lemma 5. 13. ✷ We mention a result for future use.

Lemma 5.15
The automorphism σ of Ĥq sends Proof: This is routinely checked using the action of σ given in Lemma 3.5.✷ 6 The proof of Theorems 4.1, 4.2, 4.3.
In this section we prove the first three theorems from Section 4.
(ii) Using (i) we have Therefore uv + (uv) −1 commutes with u.Similarly uv + (uv) −1 commutes with v. ✷ Corollary 6.2For distinct i, j ∈ I, (ii) t i t j + (t i t j ) −1 commutes with each of t i , t j .
Proof: By Lemma 6.1 and since t k + t −1 k is central for k ∈ I. ✷ Definition 6.3 We define elements A, B, C in Ĥq as follows: Proof: The generator τ fixes each of A, B, C by Lemma 6.4 and since τ (h) = t −1 0 ht 0 for all h ∈ Ĥq .The generator ρ sends A → B → C → A by Lemma 3.5 and Definition 6.3.Similarly the generator σ swaps A, B. Define C ′ = σ(C).We show that C ′ satisfies the equations of the lemma statement.We first show that By (30) along with (20) and ( 22), Using Lemma 5.15 along with (21) and (23), To verify (31), evaluate the left-hand side using (32)-( 34) and simplify the result using Proposition 5.14 and (13).We have verified (31).Next we show that To obtain (35), apply σ to each side of (31) and evaluate the result.To aid in this evaluation, recall that σ swaps A, B; also σ swaps C, C ′ since σ 2 = τ and τ (C) = C.By these comments and Lemma 3.6 we routinely obtain (35).✷ The following is a variation on [7, Theorem 5.2].
Proposition 6.6 In the algebra Ĥq the elements A, B, C are related as follows: Proof: To get the last equation, eliminate C ′ from the equations of Lemma 6.5.To get the other two equations use the B 3 action from Lemma 3.5.Specifically, apply ρ twice to the last equation and use the data in Lemma 3.6, together with the fact that ρ cyclically permutes A, B, C. ✷ Proof of Theorem 4.1: Immediate from Lemma 6.4 and Proposition 6.6.✷ Back in Definition 2.2 we defined some elements α, β, γ of ∆ q .From now on we retain the notation α, β, γ for their images under the map ψ : ∆ q → Ĥq .Thus the elements α, β, γ of Ĥq satisfy 7 A basis for the F-vector space Ĥq Our next general goal is to prove Theorem 4.4.The proof will be completed in Section 9.
In the present section we obtain a basis for the F-vector space Ĥq .The basis consists of We also obtain a set of relations for Ĥq called reduction rules.The reduction rules show how to write any given element of Ĥq as a linear combination of the basis elements (39).
To begin the basis project, we are going to display a presentation of Ĥq that contains detailed information about how the generators commute past each other.We will give two versions of this presentation.For version I we attempt to optimize clarity.For version II we attempt to optimize utility.We hope that taken together the two versions are reasonably clear and useful.The relations in version II become our reduction rules.
We now give version I.
Proposition 7.1 The F-algebra Ĥq has a presentation by generators Proof: Consider the relations in the proposition statement.We now show that these relations hold in Ĥq .This is clear for the relations shown in the line, so consider the 16 Proof: In Proposition 7.1 eliminate {t i } 3 i=1 using the displayed relations 2-4, and eliminate {C i } i∈I using the displayed relations 13-16.Simplify the results using the displayed relations 5-8.✷ We just gave two versions of a presentation for Ĥq .From now on we focus on version II.This version will yield our reduction rules and basis for Ĥq .
Note 7.4 Referring to the presentation of Ĥq from Proposition 7.2, consider the relations which assert that the {T i } i∈I are central.These relations can be expressed as Definition 7.5 By a reduction rule for Ĥq we mean an equation that appears in Proposition 7.2 or Note 7.4.Of these reduction rules, the last four in Proposition 7.2 are said to be of the first kind, the preceeding five are said to be of the second kind, and the rest are said to be of the third kind.
Definition 7.6 For an integer n ≥ 0, by a word of length n in Ĥq we mean a product g 1 g 2 • • • g n such that g i is a balanced generator of Ĥq for 1 ≤ i ≤ n.We interpret the word of length 0 as the multiplicative identity in Ĥq .A word is called forbidden whenever it is the left-hand side of a reduction rule.Every forbidden word has length two.A forbidden word is said to be of the first kind (resp.second kind) (resp.third kind) whenever the corresponding reduction rule is of the first kind (resp.second kind) (resp.third kind).
Definition 7.7 Let w denote a forbidden word in Ĥq , and consider the corresponding reduction rule.By a descendent of w we mean a word that appears on the right-hand side of that reduction rule.

Proposition 7.8
The following is a basis for the F-vector space Ĥq : word in Ĥq .This word is called reducible whenever there exists an integer i (2 ≤ i ≤ n) such that g i−1 g i is forbidden.A word is called irreducible whenever it is not reducible.The list (40) consists of the irreducible words in Ĥq .Let w = g 1 g 2 • • • g n denote a word in Ĥq .By an inversion in w we mean an ordered pair of integers (i, j) such that 1 ≤ i < j ≤ n and the word g i g j is forbidden.The inversion (i, j) is of the first kind (resp.second kind) (resp.third kind) whenever the forbidden word g i g j is of the first kind (resp.second kind) (resp.third kind).Let W denote the set of all words in Ĥq .We define a partial order < on W as follows.Pick any words w, w ′ in W and write w = g 1 g 2 • • • g n .We say that w dominates w ′ whenever there exists an integer i (2 ≤ i ≤ n) such that (i − 1, i) is an inversion for w, and w ′ is obtained from w by replacing g i−1 g i by one of its descendents.In this case either (i) w has more inversions of the first kind than w ′ , or (ii) w and w ′ have the same number of inversions of the first kind, but w has more inversions of the second kind than w ′ , or (iii) w and w ′ have the same number of inversions for each of the first and second kind, but w has more inversions of the third kind than w ′ .By these comments the transitive closure of the domination relation on W is a partial order on W which we denote by <.By construction < is a semigroup partial order [3, p. 181] and satisfies the descending chain condition [3, p. 179].We now relate the partial order < to our reduction rules.Let w = g 1 g 2 • • • g n denote a reducible word in Ĥq .Then there exists an integer i (2 ≤ i ≤ n) such that g i−1 g i is forbidden.There exists a reduction rule with g i−1 g i on the left-hand side; in w we eliminate g i−1 g i using this reduction rule and thereby express w as a linear combination of words, each less than w with respect to <. Therefore the reduction rules are compatible with < in the sense of Bergman [3,p. 181].In order to employ the Diamond Lemma, we must show that the ambiguities are resolvable in the sense of Bergman [3,p. 181].There are potentially two kinds of ambiguities; inclusion ambiguities and overlap ambiguities [3, p. 181].For the present example there are no inclusion ambiguities.The nontrivial overlap ambiguities are Take for example t 0 XY .The words t 0 X and XY are forbidden.Therefore t 0 XY can be reduced in two ways; we could evaluate t 0 X first or we could evaluate XY first.Either way, after a 3-step reduction we get the same resolution, which is Therefore the ambiguity t 0 XY is resolvable.The other ambiguities listed above are similarly shown to be resolvable.Their resolutions are displayed in the tables below.

Ambiguity
Resolution We conclude that every ambiguity is resolvable, so by the Diamond Lemma [3, Theorem 1.2] the irreducible words form a basis for Ĥq .The result follows.✷ In Proposition 7.8 we gave a basis for Ĥq .In Proposition 7.14 below we give a variation on this basis.
Let λ denote an indeterminate.Let F[λ, λ −1 ] denote the F-algebra of Laurent polynomials in λ that have all coefficients in F.
Lemma 7.9 The following is a basis for the F-vector space F[λ, λ −1 ]: Proof: The vectors {λ i } i∈Z form a basis for the F-vector space F[λ, λ −1 ].List the elements of this basis in the following order: List the elements of (41) in the following order: Write each element of (43) as a linear combination of (42).Consider the corresponding coefficient matrix.This matrix is upper triangular with all diagonal entries 1.The result follows.✷ For a subset S of any algebra let S denote the subalgebra generated by S.
Definition 7.10 Let T denote the following subalgebra of Ĥq : Let {λ i } 3 i=0 denote mutually commuting indeterminates.By construction the F-algebra T is commutative and generated by t ±1 0 , T 1 , T 2 , T 3 .Therefore there exists a surjective F-algebra homomorphism ϕ : Proposition 7.11 The above homomorphism ϕ is an isomorphism.Moreover, in each line below the displayed vectors form a basis for the F-vector space T: Proof: By Lemma 7.9 the following is a basis for the F-vector space F[λ ±1 0 , λ 1 , λ 2 , λ 3 ]: The homomorphism ϕ sends the vectors (48) to the vectors (46); therefore the vectors (46) span T. The vectors (46) are linearly independent by Proposition 7.8.Therefore the vectors (46) form a basis for T. Consequently ϕ is an isomorphism and (47) is a basis for T. ✷ Recall the elements α, β, γ of Ĥq from (36)-(38).By those lines α, β, γ are contained in T.More precisely, (36)-(38) show how α, β, γ look in the basis for T from (47).The elements α, β, γ look as follows in the basis for T from (46): We now consider the subalgebras X ±1 and Y ±1 of Ĥq .By Proposition 7.8 the vectors {X i } i∈Z form a basis for X ±1 and the vectors {Y i } i∈Z form a basis for Y ±1 .
Lemma 7.12 There exists an isomorphism of F-algebras F[λ ±1 ] → X ±1 that sends λ → X.There exists an isomorphism of F-algebras Proposition 7.13 The F-linear map Proof: By Proposition 7.8, Lemma 7.12, and since (46) is a basis for T. ✷ We now give a variation on the basis for Ĥq given in Proposition 7.8.

Proposition 7.14
The following is a basis for the F-vector space Ĥq : Proof: By Proposition 7.13 and since (47) is a basis for T. ✷

The coefficient matrix
Suppose we have an element of Ĥq that we wish to express as a linear combination of the vectors (40) or (52).In order to describe the result efficiently we will use the following notation.
Definition 8.1 By Proposition 7.13 each h ∈ Ĥq can be written as Moreover for i, j ∈ Z the element t ij is uniquely determined by h.We call t ij the coefficient of Y i X j in h.The coefficient matrix for h has rows and columns indexed by Z and (i, j)-entry t ij for i, j ∈ Z.We view h : A coefficient matrix has finitely many nonzero entries.When we display a coefficient matrix, any row or column not shown has all entries zero.
Example 8.2 The coefficient matrix for A is The coefficient matrix for B is Our next goal is to compute the coefficient matrix for C. In order to simplify the computation we initially work with an element θ ∈ Ĥq that is closely related to C.
where we recall γ = (q −1 t 0 + qt −1 0 )T 2 + T 1 T 3 .Lemma 8.4 In the basis (52) the element θ looks as follows: We have (t 0 t 2 ) −1 = qt 3 T 1 − qXY −1 by Lemma 5.11.We mentioned t 3 = XT 0 − Xt 0 , and the term XY −1 can be evaluated using a reduction rule from Proposition 7.2.The result follows from these observations along with Definition 8.3.✷ Lemma 8.5 The coefficient matrix for θ is Proof: Use Definition 8.3 and Lemma 8.5.✷ Lemma 8.7 The coefficient matrix for XC is Proof: First find the coefficient matrix for Xθ.To do this, in the equation (54) multiply each term on the left by X and simplify the result using the reduction rules from Proposition 7.2.This yields the coefficient matrix for Xθ.Using this coefficient matrix and (53), we routinely obtain the coefficient matrix for XC.✷ We mention two results for later use.
Proof: In the first equation of Lemma 6.6, eliminate A using A = Y + Y −1 and B using B = X + X −1 .In the resulting equation solve for X −1 C. ✷ Lemma 8.9 Given h ∈ Ĥq and v ∈ T such that hv = 0. Then h = 0 or v = 0. where We continue to compute the coefficient matrix of D. For the next step we will display the coefficient matrix for a number of elements in Ĥq .When we display these coefficient matrices we just display the (i, j) entry for −2 ≤ i, j ≤ 2, since it turns out that all the other entries are zero.Consider the element C of Ĥq .By Lemma 8.6 the coefficient matrix for C is The coefficient matrix for Y By this and since t 0 commutes with X + X −1 , the coefficient matrix for Y By Lemma 8.7 the coefficient matrix for XC is The coefficient matrix for Y XC is The coefficient matrix for Y −1 XC is By (59) the coefficient matrix for G is We now evaluate (58) using ( 60)-(66).One routinely checks that (60) times q(T 1 T 3 − γ + qt −1 0 T 2 ) minus (61) times qt −1 0 T 3 plus (62) times q −1 minus (63) times qt −1 0 T 1 plus (64) times q plus (65) times qt −2 0 plus (66) is equal to zero.Evaluating (58) in this light we find that the coefficient matrix of D is zero.Therefore D = 0 and the result follows. ✷ From now on we retain the notation Ω for its image under the map ψ : ∆ q → Ĥq .Thus the element Ω of Ĥq satisfies Ω = (q + q −1 ) 2 − (q 10 Some results concerning algebraic independence Our next general goal is to prove Theorem 4.5.The proof will be completed in Section 12.
In the present section we establish some results about algebraic independence that will be used in the proof.
Let {x i } 4 i=1 denote mutually commuting indeterminates.Motivated by the form of (36)-( 38) and (67) we consider the following elements in F[x 1 , x 2 , x 3 , x 4 ]: 11 The structure of Ĥq In this section we establish some results about Ĥq that will be used in the proof of Theorem 4.5.Recall A = Y + Y −1 and B = X + X −1 .
Lemma 11.1 The following is a basis for Y ±1 : The following is a basis for X ±1 : Proof: Combine Lemma 7.9 and Lemma 7.12.✷ Lemma 11.2The following sums are direct: For each summand a basis is given in the table below.
The following sum is direct: For each summand a basis is given in the table below.
In other words, the vectors 1, λ −1 form a basis for a complement of Proof: One checks that the vectors The following is a basis for the F-vector space T(1 − t −2 0 ): The following is a basis for a complement of T(1 − t −2 0 ) in T: Proof: By Lemma 12.2 and the first assertion of Proposition 7.11.✷ Proposition 12. 4 The following is a basis for the F-vector space A Y X B T(1 − t −2 0 ): The following is a basis for a complement of A Y X B T(1 − t −2 0 ) in A Y X B T: Proof: Use Proposition 11.4 with ν = Y X. Evaluate this using Lemma 12.3 along with the fact that {A i } i∈N is a basis for A and {B i } i∈N is a basis for B .✷ Corollary 12.5 The following is a basis for a complement of Hq in Ĥq : Proof: This follows from the first assertion of Proposition 11.3, the definition of Hq in line (78), and the last assertion of Proposition 12.4.✷ Lemma 12. 6 The following (i)-(iv) hold: (i) C ∈ Hq .
Proof: (i) From the column on the right in the table of Lemma 11.6.
(iii) By line (78), and since B commutes with everything in T. ✷ We are about to define an F-linear map φ : Hq → Hq .To define φ we give its action on the four summands in (78).As we will see, the map φ acts on the first three summands as a scalar multiple of the identity.To give the action of φ on the fourth summand, we specify what φ does to the basis for this space given in (79).
Definition 12.7 We define an F-linear map φ : Hq → Hq such that both (i) φ acts as −q −1 times the identity on (ii) for k ∈ Z and i, j, r, s, t ∈ N the map φ sends Note 12.8 The map φ is characterized as follows.Observe that φ : Hq → Hq is the unique F-linear map that sends and satisfies the following for all h ∈ Hq : Lemma 12.9 We have φ 2 = q −2 1. Moreover φ is a bijection.
Proof: The first assertion is routinely checked using the column on the right in the table of Lemma 11.6, along with Definition 12.7.The second assertion is immediate from the first.✷ Lemma 12.10 Referring to the sum in (78), for each summand U the image of U under φ is displayed in the table below.
Proof: The table is obtained using Lemma 5.10.Line (85) is routinely checked.✷ Lemma 13.2Under the map h → t 0 h − ht −1 0 the image of Ĥq is This image is contained in A, B, C, T .
Proof: The first assertion follows from Lemma 13.1.The last assertion follows from the first assertion.✷ In (81) we displayed a direct sum decomposition of Hq .For each summand we now consider the corresponding projection map.
Definition 13.3For µ ∈ {1, X, Y, C} define an F-linear map P µ : Hq → Hq such that P µ acts as the identity on A µ B T, and as 0 on the other three summands in (81).Thus P µ is the projection from Hq onto A µ B T. For h ∈ Hq we have Moreover For h ∈ Hq we now consider how the projections P µ (h) are related to the projections π ν (h) from Definition 11.5.
Lemma 13.4 Let h denote an element of Hq , and write Proof: Let A q denote the F-algebra defined by generators A, B, C, t ±1 0 , {T i } 3 i=1 and the above relations.Since these relations hold in Ĥq there exists an F-algebra homomorphism A q → Ĥq that sends each generator A, B, C, t ±1 0 , {T i } 3 i=1 of A q to the corresponding element in Ĥq .Under this homomorphism the image of A q is the subalgebra A, B, C, T of Ĥq .We show that the homomorphism is injective.To this end, we claim that the following vectors span the F-vector space A q : To prove the claim, note that the elements A, B, C of A q satisfy the defining relations for ∆ q given in Definition 2.1.Therefore there exists an F-algebra homomorphism ∆ q → A q that sends each generator A, B, C of ∆ q to the corresponding element in A q .In (8) we displayed a basis for the F-vector space ∆ q .When our homomorphism ∆ q → A q is applied to a vector in this basis, the image is contained in the span of (102).Therefore the span of (102) contains the subalgebra of A q generated by A, B, C. By construction A q is generated by A, B, C, t ±1 0 , {T i } 3 i=1 .By definition each element A, B, C of A q commutes with each element t ±1 0 , {T i } 3 i=1 of A q .By construction the span of ( 102) is closed under multiplication by each element t ±1 0 , {T i } 3 i=1 of A q .By these comments the vectors (102) span A q .The claim is proven.When we apply our homomorphism A q → Ĥq to the vectors (102), we get the basis for A, B, C, T given in Proposition 12.13.Therefore the vectors (102) form a basis for A q and our homomorphism A q → Ĥq is injective.The result follows.✷ 15 The center of Ĥq In this section we describe the center Z( Ĥq ).
Recall that the {T i } i∈I are central in Ĥq .We are going to show that {T i } i∈I generate Z( Ĥq ), provided that q is not a root of unity.In this derivation we will repeatedly use the basis for Ĥq given in Proposition 7.8.Y i X j t k 0 T ℓ 0 T r 1 T s 2 T t 3 i, j ∈ Z k ∈ {0, 1} ℓ, r, s, t ∈ N (ℓ, r, s, t) = (0, 0, 0, 0).
Proof: Use Proposition 7.8.✷ Lemma 15.3The following is a basis for a complement of K in Ĥq : By assumption h ∈ Z( Ĥq ) so Xh − hX = 0. Therefore R contains (Xh ℓ,r,s,t − h ℓ,r,s,t X)T ℓ 0 T r 1 T s 2 T t 3 . (114) The element (114) is contained in L by (108).By these comments and (110), the element (114) is contained in KL.By Lemma 8.9 the map Ĥq → L, h → hT ℓ 0 T r 1 T s 2 T t 3 is a bijection.Under this map the image of K is KL.Therefore in line (114), the expression in parenthesis is contained in K.In other words, in the notation of Definition 15.4,X h ℓ,r,s,t − h ℓ,r,s,t X = 0. (115) Expanding (115) using (113) we obtain Simplifying this using Lemma 15.7 we obtain Adjusting the indices i, j in the above sums, 0 = i,j∈Z Y i X j α i,j−1 (q 2i − 1) + i,j∈Z Y i X j t 0 (β i,j−1 q 2i − β i,j+1 ).
By ( 116), (118) and since q is not a root of unity, By (117) or (119), and since finitely many of the β ij are nonzero, Evaluating (113) using these comments we obtain h ℓ,r,s,t = α 00 ∈ F. Define h ′ = h − h ℓ,r,s,t T ℓ 0 T r 1 T s 2 T t 3 .
Second of all, S(h ′ ) is obtained from S(h) by deleting the element (ℓ, r, s, t); therefore |S(h ′ )| = |S(h)| − 1.These two comments contradict the minimality of |S(h)|.The result follows.✷ Corollary 15.10 Assume that q is not a root of unity.Then the following is a basis for the F-vector space Z( Ĥq ): Proof: The vectors (120) span Z( Ĥq ) by Theorem 15.9.The vectors (120) are linearly independent because they are included in the linearly independent set (46).✷ Corollary 15.11 Assume that q is not a root of unity.Then there exists an isomorphism of F-algebras Z( Ĥq ) → F[λ 0 , λ 1 , λ 2 , λ 3 ] that sends T i → λ i for 0 ≤ i ≤ 3.

Acknowledgments
The author thanks Kazumasa Nomura for giving this paper a close reading and offering many valuable suggestions.

Definition 2 . 1 [ 15 ,
Definition 1.2].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

Lemma 6 . 1 [ 7 ,
Lemma 3.8].Let u, v denote invertible elements in any algebra such that each of u

) Lemma 6 . 4 Lemma 6 . 5
In the algebra Ĥq the element t 0 commutes with each of A, B, C. Proof: By Corollary 6.2(ii) and Definition 6.3.✷ The following is a variation on [7, Theorem 5.1].The B 3 action on Ĥq does the following to the elements A, B, C from Definition 6.3.The generator τ fixes each of A, B, C. The generator ρ sends A → B → C → A. The generator σ swaps A, B and sends C → C ′ where

) Proof of Theorem 4 . 2 : 3 :
Without loss we may assume g = ρ or g = σ.By Lemma 2.5 the action of ρ on ∆ q cyclically permutes A, B, C. By Lemma 6.5 the action of ρ on Ĥq cyclically permutes A, B, C. By Lemma 2.5 the action of σ on ∆ q swaps A, B and fixes γ.The action of σ on Ĥq swaps A, B by Lemma 6.5.The action of σ on Ĥq fixes γ by (38) and Lemmas 3.5, 3.6.The result follows.✷Proof of Theorem 4.In each case, chase A, B, C around the diagram and use Corollary 6.2(i).✷

Definition 15 . 1 Lemma 15 . 2
Let K denote the 2-sided ideal of Ĥq generated by {T i } i∈I .ThusK = i∈I Ĥq T i .(103)The following is a basis for the F-vector space K: Example 2.19 The descendents of BA are AB, C, γ.The descendents of BC are CB, A, α.The descendents of CA are AC, B, β.The descendents of C 2 are Ω, ACB, A 2 , B 2 , displayed relations.Displayed relation 1 is from Lemma 5.2.Displayed relations 2-4 follow from Lemma 5.4.Displayed relations 5-8 are from Lemma 5.10.Displayed relations 9-12 are from Definition 5.12.Displayed relations 13-16 are from Proposition 5.14.We have shown that the relations in the proposition statement hold in Ĥq .Conversely, one routinely checks that the relations in the proposition statement imply the defining relations for Ĥq given in Lemma 5.2.✷The F-algebra Ĥq has a presentation by generators X ±1 , Y ±1 , t 0 , {T i } i∈I and relations XX