The Universal Askey-Wilson Algebra and DAHA of Type (Cv₁,C₁)

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1. Verfasser:	Terwilliger, P.
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spelling	irk-123456789-1492382019-02-20T01:27:09Z The Universal Askey-Wilson Algebra and DAHA of Type (Cv₁,C₁) Terwilliger, P. 2013 Article The Universal Askey-Wilson Algebra and DAHA of Type (Cv₁,C₁) / P. Terwilliger // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 19 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33D80; 33D45 DOI: http://dx.doi.org/10.3842/SIGMA.2013.047 http://dspace.nbuv.gov.ua/handle/123456789/149238 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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language	English
format	Article
author	Terwilliger, P.
spellingShingle	Terwilliger, P. The Universal Askey-Wilson Algebra and DAHA of Type (Cv₁,C₁) Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Terwilliger, P.
author_sort	Terwilliger, P.
title	The Universal Askey-Wilson Algebra and DAHA of Type (Cv₁,C₁)
title_short	The Universal Askey-Wilson Algebra and DAHA of Type (Cv₁,C₁)
title_full	The Universal Askey-Wilson Algebra and DAHA of Type (Cv₁,C₁)
title_fullStr	The Universal Askey-Wilson Algebra and DAHA of Type (Cv₁,C₁)
title_full_unstemmed	The Universal Askey-Wilson Algebra and DAHA of Type (Cv₁,C₁)
title_sort	universal askey-wilson algebra and daha of type (cv₁,c₁)
publisher	Інститут математики НАН України
publishDate	2013
url	http://dspace.nbuv.gov.ua/handle/123456789/149238
citation_txt	The Universal Askey-Wilson Algebra and DAHA of Type (Cv₁,C₁) / P. Terwilliger // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 19 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT terwilligerp theuniversalaskeywilsonalgebraanddahaoftypecv1c1 AT terwilligerp universalaskeywilsonalgebraanddahaoftypecv1c1
first_indexed	2025-07-12T21:39:28Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 9 (2013), 047, 40 pages The Universal Askey–Wilson Algebra and DAHA of Type (C∨ 1 , C1) Paul TERWILLIGER Department of Mathematics, University of Wisconsin, Madison, WI 53706-1388, USA E-mail: terwilli@math.wisc.edu Received December 22, 2012, in final form July 07, 2013; Published online July 15, 2013 http://dx.doi.org/10.3842/SIGMA.2013.047 Abstract. Let F denote a field, and fix a nonzero q ∈ F such that q4 6= 1. The universal Askey–Wilson algebra ∆q is the associative F-algebra defined by generators and relations in the following way. The generators are A, B, C. The relations assert that each of A+ qBC − q−1CB q2 − q−2 , B + qCA− q−1AC q2 − q−2 , C + qAB − q−1BA q2 − q−2 is central in ∆q. The universal DAHA Ĥq of type (C∨1 , C1) is the associative F-algebra defined by generators {t±1i }3i=0 and relations (i) tit −1 i = t−1i ti = 1; (ii) ti + t−1i is central; (iii) t0t1t2t3 = q−1. We display an injection of F-algebras ψ : ∆q → Ĥq that sends A 7→ t1t0 + (t1t0)−1, B 7→ t3t0 + (t3t0)−1, C 7→ t2t0 + (t2t0)−1. For the map ψ we compute the image of the three central elements mentioned above. The algebra ∆q has another central element of interest, called the Casimir element Ω. We compute the image of Ω under ψ. We describe how the Artin braid group B3 acts on ∆q and Ĥq as a group of automorphisms. We show that ψ commutes with these B3 actions. Some related results are obtained. Key words: Askey–Wilson polynomials; Askey–Wilson relations; rank one DAHA 2010 Mathematics Subject Classification: 33D80; 33D45 1 Introduction The Askey–Wilson polynomials were introduced in [2] and soon became a major topic in spe- cial 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 alge- bra AW(3) is noncommutative and infinite-dimensional. It is defined by generators and rela- tions. 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, equation (6.1)], [17, Theorem 1.5], [18, Section 4.3], [19, equations (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 Uq(sl2). mailto:terwilli@math.wisc.edu http://dx.doi.org/10.3842/SIGMA.2013.047 2 P. Terwilliger In the present paper we relate ∆q to the universal DAHA of type (C∨1 , C1) [7]. This topic can be viewed as a universal analog of a topic considered by Koornwinder [10, 11], concerning how AW(3) is related to the DAHA of type (C∨1 , C1). We view [10, 11] as groundbreaking and the main inspiration for the present paper. In Section 16 we describe in detail how our results relate to those of Koornwinder [10, 11]. We will state our main 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 q4 6= 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 A+ qBC − q−1CB q2 − q−2 , B + qCA− q−1AC q2 − q−2 , C + qAB − q−1BA q2 − q−2 (1.1) is central in ∆q. For the central elements (1.1) 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 areA,B, γ. We gave a faithful action of the modular group PSL2(Z) on ∆q as a group of automorphisms; one generator sends (A,B,C) 7→ (B,C,A) and another generator sends (A,B, γ) 7→ (B,A, γ). We showed that {AiBjCkαrβsγt \| 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−1ACB + q−2A2 + q−2B2 + q2C2 − q−1Aα− q−1Bβ − 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 Uq(sl2). To describe this relationship we use the equitable presentation for Uq(sl2) which was introduced in [8]. This equitable presentation has generators x, y±1, z and relations yy−1 = y−1y = 1, qxy − q−1yx q − q−1 = 1, qyz − q−1zy q − q−1 = 1, qzx− q−1xz q − q−1 = 1. Let a, b, c denote mutually commuting indeterminates. Let F[a±1, b±1, c±1] denote the F-algebra of Laurent polynomials in a, b, c that have all coefficients in F. In [16, Theorems 2.16, 2.18] we displayed an injection of F-algebras \ : ∆q → Uq(sl2)⊗ F[a±1, b±1, c±1] that sends A 7→ x⊗ a+ y ⊗ a−1 + xy − yx q − q−1 ⊗ bc−1, B 7→ y ⊗ b+ z ⊗ b−1 + yz − zy q − q−1 ⊗ ca−1, C 7→ z ⊗ c+ x⊗ c−1 + zx− xz q − q−1 ⊗ ab−1. The map \ sends α 7→ Λ⊗ ( a+ a−1 ) + 1⊗ ( b+ b−1 )( c+ c−1 ) , β 7→ Λ⊗ ( b+ b−1 ) + 1⊗ ( c+ c−1 )( a+ a−1 ) , γ 7→ Λ⊗ ( c+ c−1 ) + 1⊗ ( a+ a−1 )( b+ b−1 ) , where Λ = qx+ q−1y + qz − qxyz is the normalized Casimir element of Uq(sl2) [16, Section 2]. In [16, Theorem 2.17] we showed that \ sends Ω to 1⊗ ( q + q−1 )2 − Λ2 ⊗ 1− 1⊗ ( a+ a−1 )2 − 1⊗ ( b+ b−1 )2 − 1⊗ ( c+ c−1 )2 − Λ⊗ ( a+ a−1 )( b+ b−1 )( c+ c−1 ) . The Universal Askey–Wilson Algebra and DAHA of Type (C∨1 , C1) 3 We now summarize the present paper. We first show that the following is a basis for the F-vector space ∆q: AiCjBkΩ`αrβsγt, j ∈ {0, 1}, i, k, `, r, s, t ∈ N. (1.2) 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 , C1). The algebra Ĥq is a variation on an algebra Ĥ introduced in [7]. By definition Ĥq is the F-algebra with generators {t±1i }3i=0 and relations (i) tit −1 i = t−1i ti = 1; (ii) ti + t−1i is central; (iii) t0t1t2t3 = q−1. We display an injection of F-algebras ψ : ∆q → Ĥq that sends A 7→ t1t0 + (t1t0) −1, B 7→ t3t0 + (t3t0) −1, C 7→ t2t0 + (t2t0) −1. The map ψ sends α 7→ ( q−1t0 + qt−10 )( t1 + t−11 ) + ( t2 + t−12 )( t3 + t−13 ) , β 7→ ( q−1t0 + qt−10 )( t3 + t−13 ) + ( t1 + t−11 )( t2 + t−12 ) , γ 7→ ( q−1t0 + qt−10 )( t2 + t−12 ) + ( t3 + t−13 )( t1 + t−11 ) . We show that ψ sends Ω to( q + q−1 )2 − (q−1t0 + qt−10 )2 − ( t1 + t−11 )2 − (t2 + t−12 )2 − (t3 + t−13 )2 − ( q−1t0 + qt−10 )( t1 + t−11 )( t2 + t−12 )( t3 + t−13 ) . 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 (1.2) to a linearly independent set. Adapting [5, Theorem 2.6], [7, Lemma 4.2] we show how the Artin braid group B3 acts on Ĥq as a group of automorphisms. The group B3 is a homomorphic preimage of PSL2(Z), and we mentioned earlier that PSL2(Z) acts on ∆q as a group of automorphisms. Pulling back this PSL2(Z) action we get a B3 action on ∆q as a group of automorphisms. We show that ψ commutes with the B3 actions for ∆q and Ĥq. Now consider the image of ∆q under ψ. Adapting [11, Theorem 5.1] we show that the subalgebra {h ∈ Ĥq\|t0h = ht0} is generated by this image together with t0 and {ti + t−1i }3i=1. For this subalgebra we give a presentation by generators and relations. Roughly speaking, this presentation amounts to a q-analog of [13, Theorem 2.1] and a universal analog of [10, Definition 6.1, Corollary 6.3]. Under the assumption that q is not a root of unity, we show that Z(Ĥq) is generated by {ti + t−1i }3i=0 and that Z(Ĥq) is isomorphic to a polynomial algebra in four variables. Roughly speaking, this is a universal analog of [11, Theorem 5.3]. 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\|t0h = ht0} is generated by this image together with t0 and {ti + t−1i }3i=1. In Section 14 we give a presentation for this subalgebra by generators and relations. In Section 15 we describe Z(Ĥq). In Section 16 we compare our results with those of Koornwinder [10, 11]. 4 P. Terwilliger 2 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. 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 A+ qBC − q−1CB q2 − q−2 , B + qCA− q−1AC q2 − q−2 , C + qAB − q−1BA q2 − q−2 (2.1) 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 (2.1), multiply each by q + q−1 to get α, β, γ. Thus A+ qBC − q−1CB q2 − q−2 = α q + q−1 , (2.2) B + qCA− q−1AC q2 − q−2 = β q + q−1 , (2.3) C + qAB − q−1BA q2 − q−2 = γ q + q−1 . (2.4) 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 PSL2(Z) has a presentation by generators p, s and relations p3 = 1, s2 = 1. See for example [1]. Lemma 2.3 ([15, Theorem 3.1]). The group PSL2(Z) acts on ∆q as a group of automorphisms in the following way: u A B C α β γ p(u) B C A β γ α s(u) B A C + AB −BA q − q−1 β α γ The group PSL2(Z) has a central extension called the Artin braid group B3. The group B3 is defined as follows. Definition 2.4. The Artin braid group B3 is defined by generators ρ, σ and relations ρ3 = σ2. For notational convenience define τ = ρ3 = σ2. There exists a group homomorphism B3 → PSL2(Z) that sends ρ 7→ p and σ 7→ s. Via this homomorphism we pull back the PSL2(Z) action on ∆q, to get a B3 action on ∆q as a group of automorphisms. This action is described as follows. Lemma 2.5. The group B3 acts on ∆q as a group of automorphisms such that τ(h) = h for all h ∈ ∆q and ρ, σ do the following: u A B C α β γ ρ(u) B C A β γ α σ(u) B A C + AB −BA q − q−1 β α γ 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. The Universal Askey–Wilson Algebra and DAHA of Type (C∨1 , C1) 5 Definition 2.6 ([15, Lemma 6.1]). Define an element Ω ∈ ∆q by Ω = q−1ACB + q−2A2 + q−2B2 + q2C2 − q−1Aα− q−1Bβ − qCγ. (2.5) We call Ω the Casimir element of ∆q. Lemma 2.7 ([15, Theorems 6.2, 8.2]). The Casimir element Ω is contained in the center Z(∆q). Moreover {Ωiαrβsγt \| i, r, s, t ∈ N} is a basis for the F-vector space Z(∆q), provided that q is not a root of unity. Lemma 2.8 ([15, Theorem 6.4]). The Casimir element Ω is f ixed by everything in B3. Given an F-algebra A, by an antiautomorphism of A we mean an F-linear bijection ζ : A → A such that (uv)ζ = vζuζ for all u, v ∈ A. Lemma 2.9. There exists an antiautomorphism † of ∆q that sends A 7→ B, B 7→ A, C 7→ C, α 7→ β, β 7→ α, γ 7→ γ. Moreover †2 = 1. Proof. Routine using (2.2)–(2.4). � Lemma 2.10. The Casimir element Ω is fixed by the antiautomorphism † from Lemma 2.9. Proof. This follows from [15, Lemma 6.1]. � We mention how ∆q and ∆q−1 are related. Lemma 2.11. There exists an isomorphism of F-algebras ξ : ∆q → ∆q−1 that sends A 7→ B, B 7→ A, C 7→ C, α 7→ β, β 7→ α, γ 7→ γ. Proof. Routine using (2.2)–(2.4). � Lemma 2.12. The isomorphism ξ : ∆q → ∆q−1 from Lemma 2.11 sends the Casimir element of ∆q to the Casimir element of ∆q−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 BA = q2AB + q ( q2 − q−2 ) C − q ( q − q−1 ) γ, BC = q−2CB − q−1 ( q2 − q−2 ) A+ q−1 ( q − q−1 ) α, CA = q−2AC − q−1(q2 − q−2)B + q−1 ( q − q−1 ) β, C2 = q−2Ω− q−3ACB − q−4A2 − q−4B2 + q−3Aα+ q−3Bβ + q−1Cγ. Proof. Referring to the above four equations, the first three are reformulations of (2.2)–(2.4) and the fourth is a reformulation of (2.5). � 6 P. Terwilliger 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 ΩA = AΩ, ΩB = BΩ, ΩC = CΩ, αA = Aα, αB = Bα, αC = Cα, βA = Aβ, βB = Bβ, βC = Cβ, γA = Aγ, γB = Bγ, γC = Cγ, αΩ = Ωα, βΩ = Ωβ, γΩ = Ωγ, βα = αβ, γα = αγ, γβ = βγ. Definition 2.16. By a reduction rule for ∆q we mean an equation which appears in Proposi- tion 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 g1g2 · · · gn such that gi 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 reduc- tion rule. By a descendent of w we mean a word that appears on the right-hand side of that reduction rule. 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 C2 are Ω, ACB, A2, B2, Aα, Bβ, Cγ. A forbidden word of the second kind has a single descendent, obtained by interchanging its two factors. Theorem 2.20. The following is a basis for the F-vector space ∆q: AiCjBkΩ`αrβsγt, j ∈ {0, 1}, i, k, `, r, s, t ∈ N. (2.6) Proof. We invoke Bergman’s Diamond Lemma [3, Theorem 1.2]. Let g1g2 · · · gn denote a word in ∆q. This word is called reducible whenever there exists an integer i (2 ≤ i ≤ n) such that gi−1gi is forbidden. The word is called irreducible whenever it is not reducible. The list (2.6) consists of the irreducible words in ∆q. Let w = g1g2 · · · gn 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 gigj is forbidden. The inversion (i, j) is of the first kind (resp. second kind) whenever the forbidden word gigj 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 = g1g2 · · · gn. 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 gi−1gi 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 = g1g2 · · · gn denote a reducible word in ∆q. Then there exists an The Universal Askey–Wilson Algebra and DAHA of Type (C∨1 , C1) 7 integer i (2 ≤ i ≤ n) such that gi−1gi is forbidden. There exists a reduction rule with gi−1gi on the left-hand side; in w we eliminate gi−1gi 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 BCA, BC2, C2A. 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 ( q2 − q−2 ) Ω + q−6ACB − q−3 ( q4 − q−4 ) A2 − q−3 ( q4 − q−4 ) B2 + q−3 ( q3 − q−3 ) Aα+ q−3 ( q3 − q−3 ) Bβ + q−3 ( q − q−1 ) Cγ. Therefore the ambiguity BCA is resolvable. The ambiguities BC2 and C2A are similarly shown to be resolvable. The resolution of BC2 is q−6BΩ− q−7ACB2 − q−8A2B − q−8B3 + q−7ABα+ q−7B2β + q−5CBγ − q−3 ( q4 − q−4 ) AC + q−2 ( q2 − q−2 ) Cα+ q−4 ( q2 − q−2 )2 B − q−4 ( q − q−1 )( q2 − q−2 ) β and the resolution of C2A is q−6AΩ− q−7A2CB − q−8AB2 − q−8A3 + q−7A2α+ q−7ABβ + q−5ACγ − q−3 ( q4 − q−4 ) CB + q−2 ( q2 − q−2 ) Cβ + q−4 ( q2 − q−2 )2 A − q−4 ( q − q−1 )( q2 − q−2 ) α. 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 of type (C∨ 1 , C1) 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 , C1). In [7] we introduced a central extension of this algebra called the universal DAHA of type (C∨1 , C1). In the present paper we will work with a variation on this algebra. For notational convenience define a four element set I = {0, 1, 2, 3}. The following definition is a variation on [7, Definition 3.1]. Definition 3.1. Let Ĥq denote the F-algebra defined by generators {t±1i }i∈I and relations tit −1 i = t−1i ti = 1, i ∈ I, ti + t−1i is central, i ∈ I, (3.1) t0t1t2t3 = q−1. We call Ĥq the universal DAHA of type (C∨1 , C1). 8 P. Terwilliger Remark 3.2. In [7, Definition 3.1] we defined an F-algebra Ĥ by generators {t±1i }i∈I and relations (i) tit −1 i = t−1i ti = 1; (ii) ti + t−1i is central; (iii) t0t1t2t3 is central. The algebra Ĥq is a homomorphic image of Ĥ. 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: t0t1t2t3, t1t2t3t0, t2t3t0t1, t3t0t1t2. Lemma 3.4. There exists an automorphism of Ĥq that sends t0 7→ t1 7→ t2 7→ t3 7→ t0. Recall the braid group B3 from Definition 2.4. Lemma 3.5. The group B3 acts on Ĥq as a group of automorphisms such that τ(h) = t−10 ht0 for all h ∈ Ĥq and ρ, σ do the following: h t0 t1 t2 t3 ρ(h) t0 t−10 t3t0 t1 t2 σ(h) t0 t−10 t3t0 t1t2t −1 1 t1 Proof. This is routinely checked, or see [7, Lemma 4.2]. � Lemma 3.6. The B3 action on Ĥq does the following to the central elements (3.1). The gen- erator τ fixes every central element. The generators ρ, σ satisfy the table below. h t0 + t−10 t1 + t−11 t2 + t−12 t3 + t−13 ρ(h) t0 + t−10 t3 + t−13 t1 + t−11 t2 + t−12 σ(h) t0 + t−10 t3 + t−13 t2 + t−12 t1 + t−11 Proof. Use (3.1) and Lemma 3.5. � Lemma 3.7. There exists a unique antiautomorphism † of Ĥq that sends t0 7→ t0, t1 7→ t3, t2 7→ t2, t3 7→ t1. Moreover †2 = 1. Proof. Use Definition 3.1. � Lemma 3.8. There exists a unique isomorphism of F-algebras ξ : Ĥq → Ĥq−1 that sends t0 7→ t−10 , t1 7→ t−13 , t2 7→ t−12 , t3 7→ t−11 . Proof. Use Definition 3.1. � 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 A 7→ t1t0 + (t1t0) −1, B 7→ t3t0 + (t3t0) −1, C 7→ t2t0 + (t2t0) −1. The homomorphism ψ sends α 7→ ( q−1t0 + qt−10 )( t1 + t−11 ) + ( t2 + t−12 )( t3 + t−13 ) , β 7→ ( q−1t0 + qt−10 )( t3 + t−13 ) + ( t1 + t−11 )( t2 + t−12 ) , γ 7→ ( q−1t0 + qt−10 )( t2 + t−12 ) + ( t3 + t−13 )( t1 + t−11 ) . The Universal Askey–Wilson Algebra and DAHA of Type (C∨1 , C1) 9 Theorem 4.2. For all g ∈ B3 the following diagram commutes: ∆q ψ−−−−→ Ĥq g y yg ∆q −−−−→ ψ Ĥq Theorem 4.3. The following diagrams commute: ∆q ψ−−−−→ Ĥq † y y† ∆q −−−−→ ψ Ĥq ∆q ψ−−−−→ Ĥq ξ y yξ ∆q−1 −−−−→ ψ Ĥq−1 Theorem 4.4. Under the homomorphism ψ from Theorem 4.1 the image of Ω is( q + q−1 )2 − (q−1t0 + qt−10 )2 − (t1 + t−11 )2 − (t2 + t−12 )2 − (t3 + t−13 )2 − ( q−1t0 + qt−10 )( t1 + t−11 )( t2 + t−12 )( t3 + t−13 ) . (4.1) Theorem 4.5. The homomorphism ψ from Theorem 4.1 is injective. 5 Preliminaries concerning Ĥq In this section we establish some basic facts about Ĥq. These facts will be used repeatedly for the rest of the paper. Definition 5.1. For the algebra Ĥq define Ti = ti + t−1i , i ∈ I. (5.1) Note that each Ti is central in Ĥq. In Definition 3.1 we gave a presentation for Ĥq involving the generators {t±1i }i∈I. Sometimes it is convenient to use {Ti}i∈I instead of {t−1i }i∈I. In terms of the generators {ti}i∈I, {Ti}i∈I the algebra Ĥq looks as follows. Lemma 5.2. The F-algebra Ĥq has a presentation by generators {ti}i∈I, {Ti}i∈I and relations t2i = Titi − 1, i ∈ I, Ti is central, i ∈ I, t0t1t2t3 = q−1. Definition 5.3. Let X, Y denote the following elements of Ĥq: X = t3t0, Y = t0t1. (5.2) Note that each of X, Y is invertible. Lemma 5.4. For the algebra Ĥq, t1 = t−10 Y, t2 = q−1Y −1t0X −1, t3 = Xt−10 . (5.3) Moreover Ĥq is generated by X±1, Y ±1, t±10 . 10 P. Terwilliger Proof. The relations (5.3) are routinely checked using Definition 3.1 and (5.2). � In terms of the generators X±1, Y ±1, t±10 the {Ti}i∈I look as follows. Lemma 5.5. For the algebra Ĥq the following (i)–(iv) hold. (i) T0 = t0 + t−10 . (ii) T1 is equal to each of t−10 Y + Y −1t0, Y t−10 + t0Y −1. (iii) T2 is equal to each of qt−10 Y X + q−1X−1Y −1t0, qXt−10 Y + q−1Y −1t0X −1, qY Xt−10 + q−1t0X −1Y −1. (iv) T3 is equal to each of t−10 X +X−1t0, Xt−10 + t0X −1. Proof. (i) Clear. (ii) Using the equation on the left in (5.3), T1 = t1 + t−11 = t−10 Y + Y −1t0. Also T1 = Y T1Y −1 = Y ( t−10 Y + Y −1t0 ) Y −1 = Y t−10 + t0Y −1. (iii) Using the middle equation in (5.3), T2 = t−12 + t2 = qXt−10 Y + q−1Y −1t0X −1. Also T2 = X−1T2X = X−1 ( qXt−10 Y + q−1Y −1t0X −1)X = qt−10 Y X + q−1X−1Y −1t0 and T2 = Y T2Y −1 = Y ( qXt−10 Y + q−1Y −1t0X −1)Y −1 = qY Xt−10 + q−1t0X −1Y −1. (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 5.6. Consider the automorphism of Ĥq from Lemma 3.4. This automorphism sends X 7→ Y 7→ q−1X−1 7→ q−1Y −1 7→ X. Lemma 5.7. Consider the automorphisms ρ, σ of Ĥq from Lemma 3.5. The automorhism ρ sends X 7→ q−1Y −1t0X −1t0, Y 7→ X. The automorphism σ sends X 7→ t−10 Y t0, Y 7→ X. The Universal Askey–Wilson Algebra and DAHA of Type (C∨1 , C1) 11 Lemma 5.8. Recall the antiautomorphism † of Ĥq from Lemma 3.7. This map swaps X, Y . Lemma 5.9. Recall the isomorphism ξ : Ĥq → Ĥq−1 from Lemma 3.8. This map sends X 7→ Y −1 and Y 7→ X−1. We now give some relations that show how t0 commutes past the X±1, Y ±1. Lemma 5.10. The following relations hold in Ĥq: t0X = X−1t0 +XT0 − T3, (5.4) t0X −1 = Xt0 −XT0 + T3, (5.5) t0Y = Y −1t0 + Y T0 − T1, (5.6) t0Y −1 = Y t0 − Y T0 + T1. (5.7) Proof. To obtain (5.4), (5.5) replace t−10 by T0 − t0 in Lemma 5.5(iv). To obtain (5.6), (5.7) replace t−10 by T0 − t0 in Lemma 5.5(ii). � We now consider how X, Y are related. Lemma 5.11. The following relations hold in Ĥq: t0t2 = q−1t−13 T1 − q−1Y X−1, t−10 t−12 = qt1T3 − qX−1Y, (5.8) t1t3 = q−1t−10 T2 − q−2X−1Y −1, t−11 t−13 = qt2T0 − Y −1X−1, (5.9) t2t0 = q−1t−11 T3 − q−1Y −1X, t−12 t−10 = qt3T1 − qXY −1, (5.10) t3t1 = q−1t−12 T0 −XY, t−13 t−11 = qt0T2 − q2Y X. (5.11) Proof. Concerning (5.8), the equation on the left comes from q−1Y X−1 = t0t 2 1t2 = t0(T1t1 − 1)t2 = q−1t−13 T1 − t0t2. The equation on the right comes from qX−1Y = t−10 t−23 t−12 = t−10 ( T3t −1 3 − 1 ) t−12 = qt1T3 − t−10 t−12 . To obtain (5.9)–(5.11), repeatedly apply the automorphism from Lemma 3.4 to everything in (5.8), and use Lemma 5.6. � Definition 5.12. Let {Ci}i∈I denote the following elements in Ĥq: C0 = q ( qY X − q−1XY ) , C1 = − ( q−1Y X−1 − qX−1Y ) , C2 = q−1 ( qY −1X−1 − q−1X−1Y −1 ) , C3 = − ( q−1Y −1X − qXY −1 ) . Lemma 5.13. The automorphism from Lemma 3.4 sends C0 7→ C1 7→ C2 7→ C3 7→ C0. Proof. Use Lemma 5.6 and Definition 5.12. � 12 P. Terwilliger Proposition 5.14. The following relations hold in Ĥq: C0 = qT2t0 + T3t1 + q−1T0t2 + T1t3 − q−1T0T2 − T1T3, (5.12) C1 = T2t0 + qT3t1 + T0t2 + q−1T1t3 − T0T2 − q−1T1T3, (5.13) C2 = q−1T2t0 + T3t1 + qT0t2 + T1t3 − q−1T0T2 − T1T3, (5.14) C3 = T2t0 + q−1T3t1 + T0t2 + qT1t3 − T0T2 − q−1T1T3. (5.15) Proof. To verify (5.12), use (5.11) together with t−13 t−11 = (T3 − t3)(T1 − t1) = T1T3 − t3T1 − t1T3 + t3t1. To verify (5.13)–(5.15), repeatedly apply the automorphism from Lemma 3.4 to everything in (5.12), and use Lemma 5.13. � We mention a result for future use. Lemma 5.15. The automorphism σ of Ĥq sends t1t3 7→ q−1t−10 t−12 , t−13 t−11 7→ qt2t0, t0t2 7→ q−1t−13 t−11 , t−12 t−10 7→ qt1t3. 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. Lemma 6.1 ([7, Lemma 3.8]). Let u, v denote invertible elements in any algebra such that each of u+ u−1, v + v−1 is central. Then (i) uv + (uv)−1 = vu+ (vu)−1; (ii) uv + (uv)−1 commutes with each of u, v. Proof. (i) Write U = u+ u−1 and V = v + v−1. We have both uv + (uv)−1 = uv + (V − v)(U − u) = uv + vu− vU − uV + UV, vu+ (vu)−1 = vu+ (U − u)(V − v) = vu+ uv − uV − vU + UV. The result follows. (ii) Using (i) we have u−1 ( uv + v−1u−1 ) u = vu+ u−1v−1 = uv + v−1u−1. Therefore uv + (uv)−1 commutes with u. Similarly uv + (uv)−1 commutes with v. � Corollary 6.2. For distinct i, j ∈ I, (i) titj + (titj) −1 = tjti + (tjti) −1. (ii) titj + (titj) −1 commutes with each of ti, tj. Proof. By Lemma 6.1 and since tk + t−1k is central for k ∈ I. � The Universal Askey–Wilson Algebra and DAHA of Type (C∨1 , C1) 13 Definition 6.3. We define elements A, B, C in Ĥq as follows: A = t1t0 + (t1t0) −1 = t0t1 + (t0t1) −1 = Y + Y −1, B = t3t0 + (t3t0) −1 = t0t3 + (t0t3) −1 = X +X−1, C = t2t0 + (t2t0) −1 = t0t2 + (t0t2) −1. (6.1) Lemma 6.4. In the algebra Ĥq the element t0 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]. Lemma 6.5. The B3 action on Ĥq does the following to the elements A, B, C from Defini- tion 6.3. The generator τ fixes each of A, B, C. The generator ρ sends A 7→ B 7→ C 7→ A. The generator σ swaps A, B and sends C 7→ C ′ where qC + q−1C ′ +AB = q−1C + qC ′ +BA = ( q−1t0 + qt−10 ) T2 + T1T3. Proof. The generator τ fixes each of A, B, C by Lemma 6.4 and since τ(h) = t−10 ht0 for all h ∈ Ĥq. The generator ρ sends A 7→ B 7→ C 7→ 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 qC + q−1C ′ +AB = ( q−1t0 + qt−10 ) T2 + T1T3. (6.2) Since A = Y + Y −1 and B = X +X−1, AB = Y X + Y X−1 + Y −1X + Y −1X−1. (6.3) By (6.1) along with (5.8) and (5.10), C = t0t2 + t−12 t−10 = ( qt3 + q−1t−13 ) T1 − qXY −1 − q−1Y X−1. Using (6.1) and Lemma 5.15 along with (5.9) and (5.11), C ′ = qt1t3 + q−1t−13 t−11 = T0T2 − qY X − q−1X−1Y −1. (6.4) To verify (6.2), evaluate the left-hand side using (6.3)–(6.4) and simplify the result using Definition 5.12, Proposition 5.14, and (5.1). We have verified (6.2). Next we show that q−1C + qC ′ +BA = ( q−1t0 + qt−10 ) T2 + T1T3. (6.5) To obtain (6.5), apply σ to each side of (6.2) and evaluate the result. To aid in this evaluation, recall that σ swaps A, B; also σ swaps C, C ′ since σ2 = τ and τ(C) = C. By these comments and Lemma 3.6 we routinely obtain (6.5). � 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: A+ qBC − q−1CB q2 − q−2 = ( q−1t0 + qt−10 ) T1 + T2T3 q + q−1 , B + qCA− q−1AC q2 − q−2 = ( q−1t0 + qt−10 ) T3 + T1T2 q + q−1 , C + qAB − q−1BA q2 − q−2 = ( q−1t0 + qt−10 ) T2 + T3T1 q + q−1 . 14 P. Terwilliger Proof. To get the last equation, eliminate C ′ from the equations of Lemma 6.5. To get the other two equations use the B3 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 and fixes t0. � 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 α = ( q−1t0 + qt−10 ) T1 + T2T3, (6.6) β = ( q−1t0 + qt−10 ) T3 + T1T2, (6.7) γ = ( q−1t0 + qt−10 ) T2 + T3T1. (6.8) Proof of Theorem 4.2. 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 (6.8) and Lemmas 3.5, 3.6. The result follows. � Proof of Theorem 4.3. In each case, chase A, B, C around the diagram using Theorem 4.1 and Corollary 6.2(i), together with Lemma 2.9 and 3.7 for † and with Lemma 2.11 and 3.8 for ξ. � 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 Y iXjtk0T ` 0T r 1T s 2T t 3, i, j ∈ Z, k ∈ {0, 1}, `, r, s, t ∈ N. (7.1) 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 (7.1). 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 X±1, Y ±1, {ti}i∈I, {Ti}i∈I, {Ci}i∈I and relations XX−1 = 1, X−1X = 1, Y Y −1 = 1, Y −1Y = 1, the {Ti}i∈I are central, t20 = t0T0 − 1, t1 = (T0 − t0)Y, t2 = q−1Y −1t0X −1, t3 = X(T0 − t0), t0X = X−1t0 +XT0 − T3, t0X −1 = Xt0 −XT0 + T3, The Universal Askey–Wilson Algebra and DAHA of Type (C∨1 , C1) 15 t0Y = Y −1t0 + Y T0 − T1, t0Y −1 = Y t0 − Y T0 + T1, XY = q2Y X − C0, X−1Y = q−2Y X−1 + q−1C1, X−1Y −1 = q2Y −1X−1 − q2C2, XY −1 = q−2Y −1X + q−1C3, C0 = qT2t0 + T3t1 + q−1T0t2 + T1t3 − q−1T0T2 − T1T3, C1 = T2t0 + qT3t1 + T0t2 + q−1T1t3 − T0T2 − q−1T1T3, C2 = q−1T2t0 + T3t1 + qT0t2 + T1t3 − q−1T0T2 − T1T3, C3 = T2t0 + q−1T3t1 + T0t2 + qT1t3 − T0T2 − q−1T1T3. 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 displayed rela- tions. 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. � We now give version II. Roughly speaking, this version amounts to a universal analog of [10, Proposition 5.2]. Proposition 7.2. The F-algebra Ĥq has a presentation by generators X±1, Y ±1, t0, {Ti}i∈I and relations XX−1 = 1, X−1X = 1, Y Y −1 = 1, Y −1Y = 1, the {Ti}i∈I are central, t20 = t0T0 − 1, t0X = X−1t0 +XT0 − T3, t0X −1 = Xt0 −XT0 + T3, t0Y = Y −1t0 + Y T0 − T1, t0Y −1 = Y t0 − Y T0 + T1, XY = q2Y X − qt0T2 + q−1T0T2 + Y −1t0T3 − q−2Y −1T0T3 + q−2Y −1XT 2 0 − q−2Y −1Xt0T0 −XT0T1 +Xt0T1, X−1Y = q−2Y X−1 + ( q − q−1 ) q−1T1T3 − q−1T0T2 + q−1t0T2 − Y −1t0T3 + q−2XT0T1 − q−2Xt0T1 + q−2Y −1T0T3 − q−2Y −1XT 2 0 + q−2Y −1Xt0T0, X−1Y −1 = q2Y −1X−1 − q2Y −1T0T3 + q2Y −1t0T3 + qT0T2 − qt0T2 − q2XT0T1 + q2Xt0T1 + q2Y −1XT 2 0 − q2Y −1Xt0T0, XY −1 = q−2Y −1X +XT0T1 −Xt0T1 − q−2Y −1XT 2 0 + q−2Y −1Xt0T0 + q−2Y −1T0T3 − q−2Y −1t0T3 − q−1T0T2 + q−1t0T2. Proof. In Proposition 7.1 eliminate {ti}3i=1 using the displayed relations 2–4, and eliminate {Ci}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. 16 P. Terwilliger Definition 7.3. The generators X±1, Y ±1, t0, {Ti}i∈I of Ĥq are called balanced. Note 7.4. Referring to the presentation of Ĥq from Proposition 7.2, consider the relations which assert that the {Ti}i∈I are central. These relations can be expressed as TiX ±1 = X±1Ti, TiY ±1 = Y ±1Ti, Tit0 = t0Ti, i ∈ I, TiTj = TjTi, i, j ∈ I, i > j. 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 g1g2 · · · gn such that gi 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. Roughly speaking, the following result amounts to a universal analog of [10, Theorem 5.3]. Proposition 7.8. The following is a basis for the F-vector space Ĥq: Y iXjtk0T ` 0T r 1T s 2T t 3 i, j ∈ Z, k ∈ {0, 1}, `, r, s, t ∈ N. (7.2) Proof. We invoke Bergman’s Diamond Lemma [3, Theorem 1.2]. Let g1g2 · · · gn denote a word in Ĥq. This word is called reducible whenever there exists an integer i (2 ≤ i ≤ n) such that gi−1gi is forbidden. A word is called irreducible whenever it is not reducible. The list (7.2) consists of the irreducible words in Ĥq. Let w = g1g2 . . . gn 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 gigj is forbidden. The inversion (i, j) is of the first kind (resp. second kind) (resp. third kind) whenever the forbidden word gigj 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 = g1g2 · · · gn. 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 gi−1gi 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 = g1g2 · · · gn denote a reducible word in Ĥq. Then there exists an integer i (2 ≤ i ≤ n) such that gi−1gi is forbidden. There exists a reduction rule with gi−1gi on the left-hand side; in w we eliminate gi−1gi 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]. The Universal Askey–Wilson Algebra and DAHA of Type (C∨1 , C1) 17 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 t0XY, t0X −1Y, t0XY −1, t0X −1Y −1, t20X, t20X −1, t20Y, t20Y −1, XX−1Y, XX−1Y −1, X−1XY, X−1XY −1, XY Y −1, XY −1Y, X−1Y Y −1, X−1Y −1Y, t0XX −1, t0X −1X, t0Y Y −1, t0Y −1Y. Take for example t0XY . The words t0X and XY are forbidden. Therefore t0XY can be reduced in two ways; we could evaluate t0X first or we could evaluate XY first. Either way, after a 3-step reduction we get the same resolution, which is q2Y XT0 + q−2Y X−1T0 + q2Y −1X−1t0 + q2Y −1XT0 − q2XT1 + ( q−2 − 1 ) XT 2 0 T1 + ( 1− q−2 ) Xt0T0T1 −X−1T1 − Y T3 − q2Y −1T3 − ( q − q−1 ) t0T0T2 + ( 1− q−2 ) T0T1T3 + qT2. Therefore the ambiguity t0XY is resolvable. The other ambiguities listed above are similarly shown to be resolvable. Their resolutions are displayed in the tables below. Ambiguity Resolution t0X −1Y q−2Y −1Xt0 − q−2Y −1XT0 + Y T3 + q−2Y −1T3 − q−1T2, t0XY −1 q−2Y X−1t0 − q−2Y X−1T0 + (q−2 − 1)Xt0T0T1 + (1− q−2)XT 2 0 T1 + q−2XT1 +X−1T1 + (1− q−2)t0T1T3 + (q−2 − 1)T0T1T3 − q−1T2, t0X −1Y −1 q2Y Xt0 − q2Y XT0 + qT2 t20X X−1t0T0 +XT 2 0 −X − T0T3 t20X −1 Xt0T0 −XT 2 0 −X−1 + T0T3 t20Y Y −1t0T0 + Y T 2 0 − Y − T0T1 t20Y −1 Y t0T0 − Y T 2 0 − Y −1 + T0T1 Ambiguity XX−1Y XX−1Y −1 X−1XY X−1XY −1 Resolution Y Y −1 Y Y −1 Ambiguity XY Y −1 XY −1Y X−1Y Y −1 X−1Y −1Y Resolution X X X−1 X−1 Ambiguity t0XX −1 t0X −1X t0Y Y −1 t0Y −1Y Resolution t0 t0 t0 t0 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]: λk(λ+ λ−1)`, k ∈ {0, 1}, ` ∈ N. (7.3) 18 P. Terwilliger Proof. The vectors {λi}i∈Z form a basis for the F-vector space F[λ, λ−1]. List the elements of this basis in the following order: 1, λ, λ−1, λ2, λ−2, λ3, λ−3, . . . . (7.4) List the elements of (7.3) in the following order: 1, λ, λ+ λ−1, λ ( λ+ λ−1 ) , ( λ+ λ−1 )2 , λ ( λ+ λ−1 )2 , . . . . (7.5) Write each element of (7.5) as a linear combination of (7.4). 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: T = 〈t±10 , T1, T2, T3〉. Let {λi}3i=0 denote mutually commuting indeterminates. By construction the F-algebra T is commutative and generated by t±10 , T1, T2, T3. Therefore there exists a surjective F-algebra homomorphism ϕ : F[λ±10 , λ1, λ2, λ3]→ T that sends λ±10 7→ t±10 , λ1 7→ T1, λ2 7→ T2, λ3 7→ T3. 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: tk0T ` 0T r 1T s 2T t 3, k ∈ {0, 1}, `, r, s, t ∈ N; (7.6) tk0T r 1T s 2T t 3, k ∈ Z, r, s, t ∈ N. (7.7) Proof. By Lemma 7.9 the following is a basis for the F-vector space F[λ±10 , λ1, λ2, λ3]: λk0 ( λ0 + λ−10 )` λr1λ s 2λ t 3, k ∈ {0, 1}, `, r, s, t ∈ N. (7.8) The homomorphism ϕ sends the vectors (7.8) to the vectors (7.6); therefore the vectors (7.6) span T. The vectors (7.6) are linearly independent by Proposition 7.8. Therefore the vectors (7.6) form a basis for T. Consequently ϕ is an isomorphism and (7.7) is a basis for T. � Recall the elements α, β, γ of Ĥq from (6.6)–(6.8). By those equations α, β, γ are contained in T. More precisely, (6.6)–(6.8) show how α, β, γ look in the basis for T from (7.7). The elements α, β, γ look as follows in the basis for T from (7.6): α = qT0T1 − ( q − q−1 ) t0T1 + T2T3, β = qT0T3 − ( q − q−1 ) t0T3 + T1T2, γ = qT0T2 − ( q − q−1 ) t0T2 + T3T1. We now consider the subalgebras 〈X±1〉 and 〈Y ±1〉 of Ĥq. By Proposition 7.8 the vectors {Xi}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 7→ X. There exists an isomorphism of F-algebras F[λ±1]→ 〈Y ±1〉 that sends λ 7→ Y . Proposition 7.13. The F-linear map 〈Y ±1〉 ⊗ 〈X±1〉 ⊗ T→ Ĥq, u⊗ v ⊗ w 7→ uvw is a bijection. The Universal Askey–Wilson Algebra and DAHA of Type (C∨1 , C1) 19 Proof. By Proposition 7.8, Lemma 7.12, and since (7.6) 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: Y iXjtk0T r 1T s 2T t 3, i, j, k ∈ Z, r, s, t ∈ N. (7.9) Proof. By Proposition 7.13 and since (7.7) is a basis for T. � 8 The coefficient matrix Suppose we have an element of Ĥq that we wish to express as a linear combination of the vectors (7.2) or (7.9). 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 h = ∑ i,j∈Z Y iXjtij , tij ∈ T. Moreover for i, j ∈ Z the element tij is uniquely determined by h. We call tij the coefficient of Y iXj in h. The coefficient matrix for h has rows and columns indexed by Z and (i, j)-entry tij for i, j ∈ Z. We view h : · · · X−2 X−1 1 X X2 · · · ... ... Y −2 t−2,−2 t−2,−1 t−2,0 t−2,1 t−2,2 Y −1 t−1,−2 t−1,−1 t−1,0 t−1,1 t−1,2 1 · · · t0,−2 t0,−1 t0,0 t0,1 t0,2 · · · Y t1,−2 t1,−1 t1,0 t1,1 t1,2 Y 2 t2,−2 t2,−1 t2,0 t2,1 t2,2 ... ... 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 X−1 1 X Y −1 0 1 0 1 0 0 0 Y 0 1 0 The coefficient matrix for B is X−1 1 X Y −1 0 0 0 1 1 0 1 Y 0 0 0 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. 20 P. Terwilliger Definition 8.3. Define θ ∈ Ĥq such that qC = γ − θt−10 , (8.1) where we recall γ = ( q−1t0 + qt−10 ) T2 + T1T3. Lemma 8.4. In the basis (7.9) the element θ looks as follows: θ = Y X−1t0 − Y −1Xt−10 + Y −1T3 +XT1 + q−1t20T2. (8.2) Proof. Recall that C = t0t2 + (t0t2) −1. We have t0t2 = q−1t−13 T1 − q−1Y X−1 by Lemma 5.11. Also t−13 = T3 − t3 and t3 = XT0 −Xt0. By these comments t0t2 = q−1T1T3 − q−1XT0T1 + q−1Xt0T1 − q−1Y X−1. We have (t0t2) −1 = qt3T1 − qXY −1 by Lemma 5.11. We mentioned t3 = XT0 −Xt0, 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 X−1 1 X Y −1 0 T3 −t−10 1 0 q−1t20T2 T1 Y t0 0 0 Proof. Use Lemma 8.4. � Lemma 8.6. The coefficent matrix for C is X−1 1 X Y −1 0 −q−1t−10 T3 q−1t−20 1 0 t−10 T2 + q−1T1T3 −q−1t−10 T1 Y −q−1 0 0 Proof. Use Definition 8.3 and Lemma 8.5. � Lemma 8.7. The coefficient matrix for XC is X−2 X−1 1 X X2 Y −2 0 0 0 0 0 Y −1 0 q−3t0T3 −q−1T 2 3 − q−3t20 − q−3 q−2(q−1t0 + qt−10 )T3 −q−3 1 0 −q−2t0T2 q−1t0T1 + T2T3 0 0 Y 0 0 −q 0 0 Y 2 0 0 0 0 0 Proof. First find the coefficient matrix for Xθ. To do this, in the equation (8.2) 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 (8.1), we routinely obtain the coefficient matrix for XC. � We mention two results for later use. The Universal Askey–Wilson Algebra and DAHA of Type (C∨1 , C1) 21 Lemma 8.8. We have X−1C = q−2C ( X +X−1 ) −XC − q−1 ( q2 − q−2 )( Y + Y −1 ) + q−1 ( q − q−1 ) α, where we recall α = ( q−1t0 + qt−10 ) T1 + T2T3. 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−1C. � Lemma 8.9. Given h ∈ Ĥq and v ∈ T such that hv = 0. Then h = 0 or v = 0. Proof. We assume v 6= 0 and show h = 0. Following Definition 8.1 write h = ∑ i,j∈Z Y iXjtij , tij ∈ T. In this equation we multiply each term on the right by v to obtain 0 = ∑ i,j∈Z Y iXjtijv. Note that tijv ∈ T for i, j ∈ Z. By this and Proposition 7.13 we find tijv = 0 for i, j ∈ Z. The algebra T is isomorphic to F[λ±10 , λ1, λ2, λ3] by Proposition 7.11. The algebra F[λ±10 , λ1, λ2, λ3] is a domain, so T is a domain. By this and since v 6= 0 we find tij = 0 for all i, j ∈ Z. Therefore h = 0. � 9 The proof of Theorem 4.4 In this section we prove Theorem 4.4. Recall the Casimir element Ω in ∆q, from Definition 2.6. Let Ω′ denote the element (4.1), so that Ω′ = ( q + q−1 )2 − (q−1t0 + qt−10 )2 − T 2 1 − T 2 2 − T 2 3 − ( q−1t0 + qt−10 ) T1T2T3. Theorem 4.4 asserts that Ω′ is the image of Ω under ψ. Proof of Theorem 4.4. By Definition 2.6 and Theorem 4.1 the image of Ω under ψ is the following element of Ĥq: q−1ACB + q−2A2 + q−2B2 + q2C2 − q−1Aα− q−1Bβ − qCγ, (9.1) where α, β, γ are from (6.6)–(6.8). We show that (9.1) is equal to Ω′. Define D to be (9.1) minus Ω′. We show that D = 0. Our strategy is to find the coefficient matrix for D in the sense of Definition 8.1. Using A = Y + Y −1 and B = X +X−1 we obtain D = q−1 ( Y + Y −1 ) C ( X +X−1 ) + q−2 ( Y + Y −1 )2 + q−2 ( X +X−1 )2 + q2C2 − q−1 ( Y + Y −1 ) α− q−1 ( X +X−1 ) β − qCγ − Ω′. (9.2) In order to evaluate D further we consider the term C2. In this product eliminate the first factor using the formula for C from Lemma 8.6. Simplify the result using the fact that C commutes with t0; this gives C2 = q−1Y −1XCt−20 − q −1Y X−1C − q−1XCt−10 T1 − q−1Y −1Ct−10 T3 + C ( t−10 T2 + q−1T1T3 ) . 22 P. Terwilliger In the above formula we eliminate X−1C using Lemma 8.8. Evaluating (9.2) using the results we obtain D = qC ( T1T3 − γ + qt−10 T2 ) − qY −1Ct−10 T3 + q−1Y −1C ( X +X−1 ) − qXCt−10 T1 + qY XC + qY −1XCt−20 +G, (9.3) where G = q2Y 2 + q−2Y −2 − qY α− q−1Y −1α+ q−2X2 + q−2X−2 − q−1Xβ − q−1X−1β + q2 + 3q−2 − Ω′. (9.4) 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 X−2 X−1 1 X X2 Y −2 0 0 0 0 0 Y −1 0 0 −q−1t−10 T3 q−1t−20 0 1 0 0 t−10 T2 + q−1T1T3 −q−1t−10 T1 0 Y 0 −q−1 0 0 0 Y 2 0 0 0 0 0 (9.5) The coefficient matrix for Y −1C is X−2 X−1 1 X X2 Y −2 0 0 −q−1t−10 T3 q−1t−20 0 Y −1 0 0 t−10 T2 + q−1T1T3 −q−1t−10 T1 0 1 0 −q−1 0 0 0 Y 0 0 0 0 0 Y 2 0 0 0 0 0 (9.6) By this and since t0 commutes with X +X−1, the coefficient matrix for Y −1C(X +X−1) is X−2 X−1 1 X X2 Y −2 0 −q−1t−10 T3 q−1t−20 −q−1t−10 T3 q−1t−20 Y −1 0 t−10 T2 + q−1T1T3 −q−1t−10 T1 t−10 T2 + q−1T1T3 −q−1t−10 T1 1 −q−1 0 −q−1 0 0 Y 0 0 0 0 0 Y 2 0 0 0 0 0 (9.7) By Lemma 8.7 the coefficient matrix for XC is X−2 X−1 1 X X2 Y −2 0 0 0 0 0 Y −1 0 q−3t0T3 −q−1T 2 3 − q−3t20 − q−3 q−2 ( q−1t0 + qt−10 ) T3 −q−3 1 0 −q−2t0T2 q−1t0T1 + T2T3 0 0 Y 0 0 −q 0 0 Y 2 0 0 0 0 0 (9.8) The Universal Askey–Wilson Algebra and DAHA of Type (C∨1 , C1) 23 The coefficient matrix for Y XC is X−2 X−1 1 X X2 Y −2 0 0 0 0 0 Y −1 0 0 0 0 0 1 0 q−3t0T3 −q−1T 2 3 − q−3t20 − q−3 q−2 ( q−1t0 + qt−10 ) T3 −q−3 Y 0 −q−2t0T2 q−1t0T1 + T2T3 0 0 Y 2 0 0 −q 0 0 (9.9) The coefficient matrix for Y −1XC is X−2 X−1 1 X X2 Y −2 0 q−3t0T3 −q−1T 2 3 − q−3t20 − q−3 q−2 ( q−1t0 + qt−10 ) T3 −q−3 Y −1 0 −q−2t0T2 q−1t0T1 + T2T3 0 0 1 0 0 −q 0 0 Y 0 0 0 0 0 Y 2 0 0 0 0 0 (9.10) By (9.4) the coefficient matrix for G is X−2 X−1 1 X X2 Y −2 0 0 q−2 0 0 Y −1 0 0 −q−1α 0 0 1 q−2 −q−1β q2 + 3q−2 − Ω′ −q−1β q−2 Y 0 0 −qα 0 0 Y 2 0 0 q2 0 0 (9.11) We now evaluate (9.3) using (9.5)–(9.11). One routinely checks that (9.5) times q(T1T3 − γ + qt−10 T2) minus (9.6) times qt−10 T3 plus (9.7) times q−1 minus (9.8) times qt−10 T1 plus (9.9) times q plus (9.10) times qt−20 plus (9.11) is equal to zero. Evaluating (9.3) 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−1t0 + qt−10 )2 − T 2 1 − T 2 2 − T 2 3 − ( q−1t0 + qt−10 ) T1T2T3. (9.12) 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 {xi}4i=1 denote mutually commuting indeterminates. Motivated by the form of (6.6)–(6.8) and (9.12) we consider the following elements in F[x1, x2, x3, x4]: y1 = x1x2x3x4 + x21 + x22 + x23 + x24, (10.1) y2 = x1x2 + x3x4, y3 = x1x3 + x2x4, y4 = x1x4 + x2x3. (10.2) Lemma 10.1 ([16, Lemma 8.1]). The elements {yi}4i=1 in (10.1), (10.2) are algebraically inde- pendent over F. Recall the algebra T from Definition 7.10. 24 P. Terwilliger Lemma 10.2. The following are algebraically independent elements of T: Ω, α, β, γ. Proof. Recall that T is generated by t±10 , T1, T2, T3. By Proposition 7.11 the following are algebraically independent over F: t0, T1, T2, T3. Therefore the following are algebraically independent over F: q−1t0 + qt−10 , T1, T2, T3. (10.3) Denote the sequence (10.3) by {Xi}4i=1. By Lemma 10.1 the following are algebraically inde- pendent over F: X1X2X3X4 +X2 1 +X2 2 +X2 3 +X2 4 , X1X2 +X3X4, X1X3 +X2X4, X1X4 +X2X3. By (6.6)–(6.8) and (9.12) the above four elements are( q + q−1 )2 − Ω, α, β, γ. The result follows. � Definition 10.3. Let P denote the following subalgebra of T: P = 〈Ω, α, β, γ〉. We set some notation. For subspaces U , V of Ĥq define UV = SpanF{uv \|u ∈ U, v ∈ V }. In order to motivate the next few sections let us briefly return to the map ψ : ∆q → Ĥq from Theorem 4.1. Our current goal is to show that ψ is injective. Recall that ∆q is generated by A, B, C. Therefore the image of ∆q under ψ is the subalgebra 〈A,B,C〉 of Ĥq. By Theorem 2.20 the vectors (2.6) form a basis for ∆q. Applying ψ to this basis, we find that the following vectors span 〈A,B,C〉: AiCjBkΩ`αrβsγt, j ∈ {0, 1}, i, k, `, r, s, t ∈ N. (10.4) Consequently 〈A,B,C〉 = 〈A〉〈B〉P + 〈A〉C〈B〉P. (10.5) In order to show that ψ is injective, it suffices to show that the vectors (10.4) are linearly in- dependent. To show this, it will be convenient to expand our focus from the algebra 〈A,B,C〉 to the algebra 〈A,B,C,T〉 = 〈A,B,C, t±10 , T1, T2, T3〉. By (10.5), and since everything in 〈A,B,C〉 commutes with everything in T, 〈A,B,C,T〉 = 〈A〉〈B〉T + 〈A〉C〈B〉T. (10.6) We will show that the following is a basis for the F-vector space 〈A,B,C,T〉: AiCjBkt`0T r 1T s 2T t 3, j ∈ {0, 1}, ` ∈ Z, i, k, r, s, t ∈ N. It will follow from this and Lemma 10.2 that the vectors (10.4) are linearly independent. The Universal Askey–Wilson Algebra and DAHA of Type (C∨1 , C1) 25 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〉: Y kA`, k ∈ {0, 1}, ` ∈ N. The following is a basis for 〈X±1〉: XkB`, k ∈ {0, 1}, ` ∈ N. Proof. Combine Lemma 7.9 and Lemma 7.12. � Lemma 11.2. The following sums are direct: 〈Y ±1〉 = 〈A〉+ Y 〈A〉, 〈X±1〉 = 〈B〉+X〈B〉. For each summand a basis is given in the table below. subspace basis 〈A〉 Ai i ∈ N Y 〈A〉 Y Ai i ∈ N 〈B〉 Bi i ∈ N X〈B〉 XBi i ∈ N Proof. Use Lemma 11.1. � Proposition 11.3. The following sum is direct: Ĥq = 〈A〉〈B〉T + 〈A〉X〈B〉T + 〈A〉Y 〈B〉T + 〈A〉Y X〈B〉T. (11.1) For each summand a basis is given in the table below. subspace basis 〈A〉〈B〉T AiBjtk0T r 1T s 2T t 3 k ∈ Z, i, j, r, s, t ∈ N 〈A〉X〈B〉T AiXBjtk0T r 1T s 2T t 3 k ∈ Z, i, j, r, s, t ∈ N 〈A〉Y 〈B〉T AiY Bjtk0T r 1T s 2T t 3 k ∈ Z, i, j, r, s, t ∈ N 〈A〉Y X〈B〉T AiY XBjtk0T r 1T s 2T t 3 k ∈ Z, i, j, r, s, t ∈ N Proof. By Proposition 7.13, Lemma 11.2, and since (7.7) is a basis for T. � Proposition 11.4. For ν ∈ {1, X, Y, Y X} the F-linear map 〈A〉 ⊗ 〈B〉 ⊗ T→ 〈A〉ν〈B〉T, u⊗ v ⊗ w 7→ uνvw is a bijection. Proof. Use the bases displayed in the table of Proposition 11.3. � Consider the four summands in the decomposition (11.1). For each summand we now consider the corresponding projection map. 26 P. Terwilliger Definition 11.5. For ν ∈ {1, X, Y, Y X} define an F-linear map πν : Ĥq → Ĥq such that πν acts as the identity on 〈A〉ν〈B〉T, and as 0 on the other three summands in (11.1). Thus πν is the projection from Ĥq onto 〈A〉ν〈B〉T. For h ∈ Ĥq we have πν(Ah) = Aπν(h), πν(hB) = πν(h)B, πν(hv) = πν(h)v, ∀ v ∈ T. (11.2) Moreover h = π1(h) + πX(h) + πY (h) + πY X(h). Lemma 11.6. For ν ∈ {1, X, Y, Y X} the projections πν(A), πν(B), πν(C) are given in the table below. ν πν(A) πν(B) πν(C) 1 A B q−1γ − q−2t0T2 − q−1At−10 T3 X 0 0 q−1AXt−20 − q−1Xt −1 0 T1 Y 0 0 q−1Y t−10 T3 − q−1Y B Y X 0 0 q−1Y X ( 1− t−20 ) Proof. To get πν(A) and πν(B), note that each of A, B is contained in 〈A〉〈B〉T. To get πν(C), consider the formula for C from Lemma 8.6. In this formula eliminate X−1, Y −1 using X−1 = B −X and Y −1 = A− Y . � 12 The proof of Theorem 4.5 In this section we will prove Theorem 4.5. To prepare for the proof, consider the following subspace of Ĥq: H̃q = 〈A〉〈B〉T + 〈A〉X〈B〉T + 〈A〉Y 〈B〉T + 〈A〉Y X〈B〉T ( 1− t−20 ) . (12.1) Lemma 12.1. The sum in (12.1) is direct. Proof. Observe that T ( 1 − t−20 ) is contained in T, so 〈A〉Y X〈B〉T ( 1 − t−20 ) is contained in 〈A〉Y X〈B〉T. The result follows in view of Proposition 11.3. � Note that F[λ±1](1− λ−2) is an ideal in F[λ±1]. Lemma 12.2. The following sum is direct: F[λ±1] = F1 + Fλ−1 + F[λ±1] ( 1− λ−2 ) . In other words, the vectors 1, λ−1 form a basis for a complement of F[λ±1](1− λ−2) in F[λ±1]. Proof. One checks that the vectors 1, λ−1, 1− λ−2, λ ( 1− λ−2 ) , λ−1(1− λ−2), λ2(1− λ−2), λ−2(1− λ−2), . . . form a basis for F[λ±1]. � Note that T ( 1− t−20 ) is an ideal in T. Lemma 12.3. The following is a basis for the F-vector space T ( 1− t−20 ) : tk0 ( 1− t−20 ) T r1T s 2T t 3, k ∈ Z, r, s, t ∈ N. The following is a basis for a complement of T ( 1− t−20 ) in T: t−k0 T r1T s 2T t 3, k ∈ {0, 1}, r, s, t ∈ N. The Universal Askey–Wilson Algebra and DAHA of Type (C∨1 , C1) 27 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−20 ) : AiY XBjtk0 ( 1− t−20 ) T r1T s 2T t 3, k ∈ Z, i, j, r, s, t ∈ N. (12.2) The following is a basis for a complement of 〈A〉Y X〈B〉T ( 1− t−20 ) in 〈A〉Y X〈B〉T: AiY XBjt−k0 T r1T s 2T t 3, k ∈ {0, 1}, i, j, r, s, t ∈ N. Proof. Use Proposition 11.4 with ν = Y X. Evaluate this using Lemma 12.3 along with the fact that {Ai}i∈N is a basis for 〈A〉 and {Bi}i∈N is a basis for 〈B〉. � Corollary 12.5. The following is a basis for a complement of H̃q in Ĥq: AiY XBjt−k0 T r1T s 2T t 3, k ∈ {0, 1}, i, j, r, s, t ∈ N. Proof. This follows from the first assertion of Proposition 11.3, the definition of H̃q in equa- tion (12.1), and the last assertion of Proposition 12.4. � Lemma 12.6. The following (i)–(iv) hold: (i) C ∈ H̃q. (ii) AH̃q ⊆ H̃q. (iii) H̃qB ⊆ H̃q. (iv) H̃qT ⊆ H̃q. Proof. (i) From the column on the right in the table of Lemma 11.6. (ii), (iv) By equation (12.1). (iii) By equation (12.1), and since B commutes with everything in T. � We are about to define an F-linear map φ : H̃q → H̃q. To define φ we give its action on the four summands in (12.1). 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 (12.2). Definition 12.7. We define an F-linear map φ : H̃q → H̃q such that both (i) φ acts as −q−1 times the identity on 〈A〉〈B〉T + 〈A〉X〈B〉T + 〈A〉Y 〈B〉T; (ii) for k ∈ Z and i, j, r, s, t ∈ N the map φ sends AiY XBjtk0 ( 1− t−20 ) T r1T s 2T t 3 7→ AiCBjtk0T r 1T s 2T t 3. Note 12.8. The map φ is characterized as follows. Observe that φ : H̃q → H̃q is the unique F-linear map that sends 1 7→ −q−1, X 7→ −q−1X, Y 7→ −q−1Y, Y X ( 1− t−20 ) 7→ C and satisfies the following for all h ∈ H̃q: φ(Ah) = Aφ(h), φ(hB) = φ(h)B, φ(hu) = φ(h)u, ∀u ∈ T. 28 P. Terwilliger Lemma 12.9. We have φ2 = q−21. 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 (12.1), for each summand U the image of U under φ is displayed in the table below. U image of U under φ 〈A〉〈B〉T 〈A〉〈B〉T 〈A〉X〈B〉T 〈A〉X〈B〉T 〈A〉Y 〈B〉T 〈A〉Y 〈B〉T 〈A〉Y X〈B〉T ( 1− t−20 ) 〈A〉C〈B〉T Proof. Use Definition 12.7. � Proposition 12.11. The following sum is direct: H̃q = 〈A〉〈B〉T + 〈A〉X〈B〉T + 〈A〉Y 〈B〉T + 〈A〉C〈B〉T. (12.3) Moreover the following is a basis for the F-vector space 〈A〉C〈B〉T: AiCBjtk0T r 1T s 2T t 3, k ∈ Z, i, j, r, s, t ∈ N. (12.4) Proof. The first assertion is a consequence of Lemma 12.1 and Lemma 12.10, together with the fact that φ is a bijection. The second assertion follows from Definition 12.7(ii) and the fact that φ is a bijection. � Proposition 12.12. The sum (10.6) is direct. Proof. The two summands in (10.6) are included among the four summands in the direct sum (12.3). � Roughly speaking, the following result amounts to a universal analog of [11, Theorem 2.6]. Proposition 12.13. The following is a basis for the F-vector space 〈A,B,C,T〉: AiCjBkt`0T r 1T s 2T t 3, j ∈ {0, 1}, ` ∈ Z, i, k, r, s, t ∈ N. (12.5) Proof. The set of vectors (12.5) consists of the basis for 〈A〉〈B〉T from the table of Propo- sition 11.3, together with the basis for 〈A〉C〈B〉T from (12.4). The result follows in view of Proposition 12.12. � Proof of Theorem 4.5. By Theorem 2.20 the vectors (2.6) form a basis for ∆q. Applying ψ to this basis, we obtain the following vectors in Ĥq: AiCjBkΩ`αrβsγt, j ∈ {0, 1}, i, k, `, r, s, t ∈ N. These vectors are linearly independent by Lemma 10.2 and since the vectors (12.5) are linearly independent. Therefore ψ is injective. � The Universal Askey–Wilson Algebra and DAHA of Type (C∨1 , C1) 29 13 The elements in Ĥq that commute with t0 We have now proven the five theorems from Section 4. Recall that these theorems describe the map ψ : ∆q → Ĥq. Our goal for the remainder of the paper is to obtain three extra results about Ĥq; these results help to illuminate ψ and may be of independent interest. The first extra result concerns the subalgebra 〈A,B,C,T〉 of Ĥq. This subalgebra was first mentioned at the end of Section 10, and a basis for it was given in Proposition 12.13. Our goal for the present section is to show that 〈A,B,C,T〉 = { h ∈ Ĥq \| t0h = ht0 } . We will be discussing the F-linear map Ĥq → Ĥq, h 7→ t0h− ht−10 . Lemma 13.1. For ν ∈ {1, X, Y, Y X} the element t0ν − νt−10 is given in the table below. ν t0ν − νt−10 1 t0 − t−10 X Bt0 − T3 Y At0 − T1 Y X q(Ct0 − T2) + (AB − T1T3)t0 Moreover (AB − T1T3)t0 = A(Bt0 − T3) + (At0 − T1)t0T3 −At0 ( t0 − t−10 ) T3. (13.1) Proof. The table is obtained using Lemma 5.10. Equation (13.1) is routinely checked. � Lemma 13.2. Under the map h 7→ t0h− ht−10 the image of Ĥq is 〈A〉 ( t0−t−10 ) 〈B〉T + 〈A〉 ( A−t−10 T1 ) 〈B〉T + 〈A〉 ( B−t−10 T3 ) 〈B〉T + 〈A〉 ( C−t−10 T2 ) 〈B〉T. 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 (12.3) we displayed a direct sum decomposition of H̃q. For each summand we now consider the corresponding projection map. Definition 13.3. For µ ∈ {1, X, Y, C} define an F-linear map Pµ : H̃q → H̃q such that Pµ acts as the identity on 〈A〉µ〈B〉T, and as 0 on the other three summands in (12.3). Thus Pµ is the projection from H̃q onto 〈A〉µ〈B〉T. For h ∈ H̃q we have Pµ(Ah) = APµ(h), Pµ(hB) = Pµ(h)B, Pµ(hv) = Pµ(h)v, ∀ v ∈ T. (13.2) Moreover h = P1(h) + PX(h) + PY (h) + PC(h). For h ∈ H̃q we now consider how the projections Pµ(h) are related to the projections πν(h) from Definition 11.5. 30 P. Terwilliger Lemma 13.4. Let h denote an element of H̃q, and write PC(h) = ∑ i,j∈N AiCBjtij , tij ∈ T. (13.3) Then π1(h) = P1(h) + ∑ i,j∈N Aiπ1(C)Bjtij , πX(h) = PX(h) + ∑ i,j∈N AiπX(C)Bjtij , πY (h) = PY (h) + ∑ i,j∈N AiπY (C)Bjtij , πY X(h) = q−1 ∑ i,j∈N AiY XBjtij ( 1− t−20 ) . Proof. We have both h = π1(h) + πX(h) + πY (h) + πY X(h), (13.4) h = P1(h) + PX(h) + PY (h) + PC(h). (13.5) In (13.5) eliminate PC(h) using (13.3), and evaluate the result using C = π1(C) + πX(C) + πY (C) + πY X(C). By Lemma 11.6 we have πY X(C) = q−1Y X ( 1− t−20 ) . Subtracting (13.5) from (13.4) and using the above comments, we obtain 0 = π1(h)− P1(h)− ∑ i,j∈N Aiπ1(C)Bjtij (13.6) + πX(h)− PX(h)− ∑ i,j∈N AiπX(C)Bjtij (13.7) + πY (h)− PY (h)− ∑ i,j∈N AiπY (C)Bjtij (13.8) + πY X(h)− q−1 ∑ i,j∈N AiY XBjtij ( 1− t−20 ) . (13.9) The elements (13.6), (13.7), (13.8), (13.9) are contained in the subspaces 〈A〉〈B〉T, 〈A〉X〈B〉T, 〈A〉Y 〈B〉T, 〈A〉Y X〈B〉T respectively. The sum of these subspaces is direct, so each of (13.6), (13.7), (13.8), (13.9) is zero. The result follows. � Lemma 13.5. For h ∈ Ĥq the following are equivalent: (i) h ∈ 〈A,B,C,T〉. (ii) h ( t0 − t−10 ) ∈ 〈A,B,C,T〉. The Universal Askey–Wilson Algebra and DAHA of Type (C∨1 , C1) 31 Proof. (i)⇒ (ii) Since t0 − t−10 ∈ T. (ii)⇒ (i) Observe by (12.1) that h ( t0 − t−10 ) ∈ H̃q. Write PC ( h ( t0 − t−10 )) = ∑ i,j∈N AiCBjtij , tij ∈ T. We first show that h ∈ H̃q. Comparing (10.6) and (12.3) we find PX ( h ( t0 − t−10 )) = 0, PY ( h ( t0 − t−10 )) = 0. (13.10) By the equation on the right in (11.2), πν ( h ( t0 − t−10 )) = πν(h) ( t0 − t−10 ) , ν ∈ {1, X, Y, Y X}. By this and Lemma 13.4, πY X(h) ( t0 − t−10 ) = πY X ( h ( t0 − t−10 )) = q−1 ∑ i,j∈N AiY XBjtij ( 1− t−20 ) . By this and Lemma 8.9, πY X(h) = q−1 ∑ i,j∈N AiY XBjtijt −1 0 . (13.11) In order to show that h ∈ H̃q we show that t0 − t−10 divides tij for all i, j ∈ N. Observe πY (h) ( t0 − t−10 ) = πY ( h ( t0 − t−10 )) by Lemma 13.4 = ∑ i,j∈N AiπY (C)Bjtij by Lemma 11.6 = q−1 ∑ i,j∈N Ai ( Y t−10 T3 − Y B ) Bjtij = q−1 ∑ r,s∈N ArY Bs ( trst −1 0 T3 − tr,s−1 ) , where tr,−1 = 0 for r ∈ N. From this we see that t0− t−10 divides trst −1 0 T3− tr,s−1 for all r, s ∈ N. By this and induction on s we find t0 − t−10 divides trs for all r, s ∈ N. In other words, for all r, s ∈ N there exists t′rs ∈ T such that trs = t′rs ( t0 − t−10 ) . Now using (13.11), πY X(h) = q−1 ∑ i,j∈N AiY XBjt′ij ( 1− t−20 ) ∈ 〈A〉Y X〈B〉T ( 1− t−20 ) . By this and (12.1) we find h ∈ H̃q. By the equation on the right in (13.2) and the equation on the left in (13.10), PX(h) ( t0 − t−10 ) = PX ( h ( t0 − t−10 )) = 0. Therefore PX(h) = 0 in view of Lemma 8.9. Similarly PY (h) ( t0 − t−10 ) = PY ( h ( t0 − t−10 )) = 0, so PY (h) = 0. Now h = P1(h) + PX(h) + PY (h) + PC(h) = P1(h) + PC(h) ∈ 〈A〉〈B〉T + 〈A〉C〈B〉T = 〈A,B,C,T〉. The result follows. � 32 P. Terwilliger Roughly speaking, the following result amounts to a universal analog of [11, Theorem 5.1]. Theorem 13.6. We have 〈A,B,C,T〉 = { h ∈ Ĥq \| t0h = ht0 } . (13.12) Proof. In (13.12) the inclusion ⊆ holds by Lemma 6.4 and since {Ti}3i=1 are central in Ĥq. We now obtain the inclusion ⊇. Pick h ∈ Ĥq such that t0h = ht0. We show h ∈ 〈A,B,C,T〉. By assumption t0h = ht0 so h ( t0 − t−10 ) = t0h − ht−10 . By Lemma 13.2 t0h − ht−10 ∈ 〈A,B,C,T〉. By these comments h ( t0 − t−10 ) ∈ 〈A,B,C,T〉. By this and Lemma 13.5 h ∈ 〈A,B,C,T〉. � 14 A presentation for the algebra 〈A,B,C,T〉 We continue to discuss the subalgebra 〈A,B,C,T〉 of Ĥq. In this section we give a presentation for 〈A,B,C,T〉 by generators and relations. Roughly speaking, this presentation amounts to a q-analog of [13, Theorem 2.1] and a universal analog of [10, Definition 6.1, Corollary 6.3]. Theorem 14.1. The F-algebra 〈A,B,C,T〉 is presented by generators and relations in the fol- lowing way. The generators are A, B, C, t±10 , {Ti}3i=1. The relations assert that each of t±10 , {Ti}3i=1 is central and t0t −1 0 = 1, t−10 t0 = 1, A+ qBC − q−1CB q2 − q−2 = α q + q−1 , B + qCA− q−1AC q2 − q−2 = β q + q−1 , C + qAB − q−1BA q2 − q−2 = γ q + q−1 , q−1ACB + q−2A2 + q−2B2 + q2C2 − q−1Aα− q−1Bβ − qCγ = ( q + q−1 )2 − (q−1t0 + qt−10 )2 − T 2 1 − T 2 2 − T 2 3 − ( q−1t0 + qt−10 ) T1T2T3, where α = ( q−1t0 + qt−10 ) T1 + T2T3, β = ( q−1t0 + qt−10 ) T3 + T1T2, γ = ( q−1t0 + qt−10 ) T2 + T3T1. Proof. Let Aq denote the F-algebra defined by generators A, B, C, t±10 , {Ti}3i=1 and the above relations. Since these relations hold in Ĥq there exists an F-algebra homomorphism Aq → Ĥq that sends each generator A, B, C, t±10 , {Ti}3i=1 of Aq to the corresponding element in Ĥq. Under this homomorphism the image of Aq 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 Aq: AiCjBkt`0T r 1T s 2T t 3, j ∈ {0, 1}, ` ∈ Z, i, k, r, s, t ∈ N. (14.1) To prove the claim, note that the elements A, B, C of Aq satisfy the defining relations for ∆q given in Definition 2.1. Therefore there exists an F-algebra homomorphism ∆q → Aq that sends each generator A, B, C of ∆q to the corresponding element in Aq. In (2.6) we displayed a basis for the F-vector space ∆q. When our homomorphism ∆q → Aq is applied to a vector in this basis, the image is contained in the span of (14.1). Therefore the span of (14.1) contains the subalgebra of Aq generated by A, B, C. By construction Aq is generated by A, B, C, t±10 , {Ti}3i=1. By definition each element A, B, C of Aq commutes with each element t±10 , {Ti}3i=1 The Universal Askey–Wilson Algebra and DAHA of Type (C∨1 , C1) 33 of Aq. By construction the span of (14.1) is closed under multiplication by each element t±10 , {Ti}3i=1 of Aq. By these comments the vectors (14.1) span Aq. The claim is proven. When we apply our homomorphism Aq → Ĥq to the vectors (14.1), we get the basis for 〈A,B,C,T〉 given in Proposition 12.13. Therefore the vectors (14.1) form a basis for Aq and our homomorphism Aq → Ĥq is injective. The result follows. � 15 The center of Ĥq In this section we describe the center Z(Ĥq). Recall that the {Ti}i∈I are central in Ĥq. We are going to show that {Ti}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. Definition 15.1. Let K denote the 2-sided ideal of Ĥq generated by {Ti}i∈I. Thus K = ∑ i∈I ĤqTi. Lemma 15.2. The following is a basis for the F-vector space K: Y iXjtk0T ` 0T r 1T s 2T t 3, i, j ∈ Z, k ∈ {0, 1}, `, r, s, t ∈ N, (`, r, s, t) 6= (0, 0, 0, 0). Proof. Use Proposition 7.8. � Lemma 15.3. The following is a basis for a complement of K in Ĥq: Y iXjtk0, i, j ∈ Z, k ∈ {0, 1}. Proof. Compare Proposition 7.8 and Lemma 15.2. � Definition 15.4. Let Hq denote the quotient F-algebra Hq = Ĥq/K. Recall that the canonical map Ĥq → Hq is a surjective F-algebra homomorphism with kernel K. For h ∈ Ĥq let h denote the image of h under this map. By construction T i = 0 for i ∈ I. Lemma 15.5. The following is a basis for the F-vector space Hq: Y i X j t k 0, i, j ∈ Z, k ∈ {0, 1}. Proof. Use Lemma 15.3. � Lemma 15.6. Referring to Definition 5.12 we have Ci = 0 for i ∈ I. Proof. By Proposition 5.14 and since T j = 0 for j ∈ I. � Lemma 15.7. The following relations hold in Hq: X Y = q2Y X, t 2 0 = −1, (15.1) t0X = X −1 t0, t0Y = Y −1 t0. (15.2) Proof. The equation on the left in (15.1) follows from Definition 5.12 and Lemma 15.6. To get the equation on the right in (15.1), apply the map h 7→ h to each side of t20 = t0T0 − 1. To get the equations in (15.2), apply the map h 7→ h to each side of (5.4) and (5.6). � 34 P. Terwilliger Definition 15.8. We endow the set N4 with a partial order ≤ as follows. Let (n0, n1, n2, n3) and (n′0, n ′ 1, n ′ 2, n ′ 3) denote elements of N4. Then (n0, n1, n2, n3) ≤ (n′0, n ′ 1, n ′ 2, n ′ 3) whenever ni ≤ n′i for 0 ≤ i ≤ 3. We have some comments. Fix (`, r, s, t) ∈ N4 and define L = ĤqT ` 0T r 1T s 2T t 3. (15.3) Then L is a 2-sided ideal of Ĥq with basis Y iXjtk0T `′ 0 T r′ 1 T s′ 2 T t′ 3 , i, j ∈ Z, k ∈ {0, 1}, (`′, r′, s′, t′) ∈ N4, (`, r, s, t) ≤ (`′, r′, s′, t′). Observe that KL is a 2-sided ideal of Ĥq with basis Y iXjtk0T `′ 0 T r′ 1 T s′ 2 T t′ 3 , i, j ∈ Z, k ∈ {0, 1}, (`′, r′, s′, t′) ∈ N4, (`, r, s, t) < (`′, r′, s′, t′). Define R = ĤqT `+1 0 + ĤqT r+1 1 + ĤqT s+1 2 + ĤqT t+1 3 . (15.4) Then R is a 2-sided ideal of Ĥq with basis Y iXjtk0T `′ 0 T r′ 1 T s′ 2 T t′ 3 , i, j ∈ Z, k ∈ {0, 1}, (`′, r′, s′, t′) ∈ N4, (`′, r′, s′, t′) 6≤ (`, r, s, t). Comparing the above bases we find L ∩R = KL. (15.5) Theorem 15.9. Assume that q is not a root of unity. Then the F-algebra Z(Ĥq) is generated by {Ti}i∈I. Proof. Consider the subalgebra 〈T0, T1, T2, T3〉 of Ĥq. This subalgebra is contained in Z(Ĥq). We assume that the containment is proper, and obtain a contradiction. Pick h ∈ Z ( Ĥq ) , h 6∈ 〈T0, T1, T2, T3〉. (15.6) In view of Proposition 7.8 we write h = ∑ `,r,s,t∈N h`,r,s,tT ` 0T r 1T s 2T t 3, h`,r,s,t ∈ 〈Y ±1〉〈X±1〉+ 〈Y ±1〉〈X±1〉t0. Define the set S(h) = { (`, r, s, t) ∈ N4 \| h`,r,s,t 6= 0 } . By construction the cardinality \|S(h)\| is finite. Without loss of generality, we assume that h has been chosen such that \|S(h)\| is minimal subject to (15.6). Note that h 6= 0 so S(h) is nonempty. There exists an element of S(h) that is not greater than any other element of S(h), with respect to the partial order ≤ from Definition 15.8. Denote this element by (`, r, s, t). We will be discussing the corresponding ideals L, R of Ĥq from (15.3) and (15.4). By construction h− h`,r,s,tT `0T r1T s2T t3 ∈ R. (15.7) The Universal Askey–Wilson Algebra and DAHA of Type (C∨1 , C1) 35 Write h`,r,s,t = ∑ i,j∈Z αijY iXj + ∑ i,j∈Z βijY iXjt0, αij , βij ∈ F. (15.8) We take the commutator of (15.7) with each of X, Y . We start with X. The ideal R contains Xh− hX − (Xh`,r,s,t − h`,r,s,tX)T `0T r 1T s 2T t 3. By assumption h ∈ Z(Ĥq) so Xh− hX = 0. Therefore R contains (Xh`,r,s,t − h`,r,s,tX)T `0T r 1T s 2T t 3. (15.9) The element (15.9) is contained in L by (15.3). By these comments and (15.5), the element (15.9) is contained in KL. By Lemma 8.9 the map Ĥq → L, g 7→ gT `0T r 1T s 2T t 3 is a bijection. Under this map the image of K is KL. Therefore in (15.9), the expression in parenthesis is contained in K. In other words, in the notation of Definition 15.4, X h`,r,s,t − h`,r,s,tX = 0. (15.10) Expanding (15.10) using (15.8) we obtain 0 = ∑ i,j∈Z αij ( X Y i X j − Y i X j X ) + ∑ i,j∈Z βij ( X Y i X j t0 − Y i X j t0X ) . Simplifying this using Lemma 15.7 we obtain 0 = ∑ i,j∈Z αijY i X j+1( q2i − 1 ) + ∑ i,j∈Z βij ( Y i X j+1 t0q 2i − Y i X j−1 t0 ) . Adjusting the indices i, j in the above sums, 0 = ∑ i,j∈Z Y i X j αi,j−1 ( q2i − 1 ) + ∑ i,j∈Z Y i X j t0 ( βi,j−1q 2i − βi,j+1 ) . By this and Lemma 15.5 we find αi,j−1 ( q2i − 1 ) = 0, i, j ∈ Z, (15.11) βi,j−1q 2i − βi,j+1 = 0, i, j ∈ Z. (15.12) Taking the commutator of (15.7) with Y , we similarly obtain αi−1,j ( q2j − 1 ) = 0, i, j ∈ Z, (15.13) βi−1,j − βi+1,jq −2j = 0, i, j ∈ Z. (15.14) By (15.11), (15.13) and since q is not a root of unity, αij = 0 if (i, j) 6= (0, 0), i, j ∈ Z. By (15.12) or (15.14), and since finitely many of the βij are nonzero, βij = 0, i, j ∈ Z. Evaluating (15.8) using these comments we obtain h`,r,s,t = α00 ∈ F. Define h′ = h− h`,r,s,tT `0T r1T s2T t3. 36 P. Terwilliger We have two comments about h′. First of all, h− h′ ∈ 〈T0, T1, T2, T3〉 ⊆ Z(Ĥq), so h′ ∈ Z(Ĥq), h′ 6∈ 〈T0, T1, T2, T3〉. Second of all, S(h′) is obtained from S(h) by deleting the element (`, r, s, t); therefore \|S(h′)\| = \|S(h)\| − 1. These two comments contradict the minimality of \|S(h)\|. The result follows. � Roughly speaking, the following two corollaries amount to a universal analog of [11, Theo- rem 5.3]. 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): T `0T r 1T s 2T t 3, `, r, s, t ∈ N. (15.15) Proof. The vectors (15.15) span Z(Ĥq) by Theorem 15.9. The vectors (15.15) are linearly independent because they are included in the linearly independent set (7.6). � Corollary 15.11. Assume that q is not a root of unity. Then there exists an isomorphism of F-algebras Z(Ĥq)→ F[λ0, λ1, λ2, λ3] that sends Ti 7→ λi for 0 ≤ i ≤ 3. Proof. Immediate from Corollary 15.10. � 16 Discussion In this section we compare our main results with the results of Koornwinder [10, 11]. Recall from Definition 3.1 that Ĥq is the universal DAHA of type (C∨1 , C1). In [10, 11] Koornwinder works with a related algebra H̃ called the DAHA of type (C∨1 , C1). We will compare these algebras shortly. Recall the set I = {0, 1, 2, 3}. By Lemma 5.2 the F-algebra Ĥq has a presentation by generators {ti}i∈I, {Ti}i∈I and relations t2i = Titi − 1, i ∈ I, Ti is central, i ∈ I, t0t1t2t3 = q−1. Definition 16.1. Let {Pi}i∈I denote scalars in F. Define an F-algebra Ĥq(P0, P1, P2, P3) by generators {ti}i∈I and relations t2i = Piti − 1, i ∈ I, t0t1t2t3 = q−1. Lemma 16.2. For i ∈ I the element ti of Ĥq(P0, P1, P2, P3) is invertible and ti + t−1i = Pi. By construction there exists a unique F-algebra homomorphism Ĥq → Ĥq(P0, P1, P2, P3) that sends ti 7→ ti and Ti 7→ Pi for i ∈ I. This map is surjective. We denote this map by ε(P0, P1, P2, P3). Recall the elements A, B, C of Ĥq: A = t1t0 + (t1t0) −1 = t0t1 + (t0t1) −1, B = t3t0 + (t3t0) −1 = t0t3 + (t0t3) −1, C = t2t0 + (t2t0) −1 = t0t2 + (t0t2) −1. We retain the notation A, B, C for their images under ε(P0, P1, P2, P3). Recall from Defini- tion 7.10 the subalgebra T = 〈t±10 , T1, T2, T3〉 of Ĥq. By (5.1), T = 〈t0, T0, T1, T2, T3〉. The subalgebra 〈A,B,C,T〉 of Ĥq was discussed in Propositions 12.12, 12.13 and Theorems 13.6, 14.1. The Universal Askey–Wilson Algebra and DAHA of Type (C∨1 , C1) 37 Definition 16.3. Consider the subalgebra 〈A,B,C,T〉 of Ĥq. Let A denote the image of 〈A,B,C,T〉 under the map ε(P0, P1, P2, P3). Thus A is the subalgebra of Ĥq(P0, P1, P2, P3) generated by A, B, C, t0. Proposition 16.4. The F-algebra A is presented by generators and relations in the following way. The generators are A, B, C, t0. The relations assert that t0 is central and t20 = P0t0 − 1, A+ qBC − q−1CB q2 − q−2 = α q + q−1 , B + qCA− q−1AC q2 − q−2 = β q + q−1 , C + qAB − q−1BA q2 − q−2 = γ q + q−1 , q−1ACB + q−2A2 + q−2B2 + q2C2 − q−1Aα− q−1Bβ − qCγ = ( q + q−1 )2 − (q−1t0 + qt−10 )2 − P 2 1 − P 2 2 − P 2 3 − ( q−1t0 + qt−10 ) P1P2P3, where α = ( q−1t0 + qt−10 ) P1 + P2P3, β = ( q−1t0 + qt−10 ) P3 + P1P2, γ = ( q−1t0 + qt−10 ) P2 + P3P1, t−10 = P0 − t0. Proof. In the relations of Theorem 14.1, first replace t−10 by T0 − t0 and then replace Ti by Pi for i ∈ I. � By the first three displayed relations in Proposition 16.4, the F-algebra A is generated by t0 together with any two of A, B, C. We now give a presentation of A by generators and relations, using the generators A, B, t0. Proposition 16.5. The F-algebra A is presented by generators A, B, t0 and relations t0A = At0, t0B = Bt0, t20 = P0t0 − 1, A2B − ( q2 + q−2 ) ABA+BA2 + ( q2 − q−2 )2 B + ( q − q−1 )2 Aγ = ( q − q−1 )( q2 − q−2 ) β, B2A− ( q2 + q−2 ) BAB +AB2 + ( q2 − q−2 )2 A+ ( q − q−1 )2 Bγ = ( q − q−1 )( q2 − q−2 ) α, q−1ACB + q−2A2 + q−2B2 + q2C2 − q−1Aα− q−1Bβ − qCγ = ( q + q−1 )2 − (q−1t0 + qt−10 )2 − P 2 1 − P 2 2 − P 2 3 − ( q−1t0 + qt−10 ) P1P2P3, where C = γ q + q−1 − qAB − q−1BA q2 − q−2 , α = ( q−1t0 + qt−10 ) P1 + P2P3, β = ( q−1t0 + qt−10 ) P3 + P1P2, γ = ( q−1t0 + qt−10 ) P2 + P3P1, t−10 = P0 − t0. Proof. In the first two displayed relations of Proposition 16.4, eliminate C using the third displayed relation. � We now bring in the work of Koornwinder [10, 11]. In [10, equations (3.1)–(3.4)] Koornwinder defines an algebra H̃. The definition involves some scalars q, a, b, c, d. For notational convenience we replace Koornwinder’s q, a, b, c, d by their squares. 38 P. Terwilliger Definition 16.6 ([10, equations (3.1)–(3.4)]). Fix nonzero scalars a, b, c, d in F. The F-algebra H̃ = H̃q(a, b, c, d) is defined by generators Z, Z−1, T1, T0 and relations( T1 + a2b2 ) (T1 + 1) = 0, ( T0 + q−2c2d2 ) (T0 + 1) = 0,( T1Z + a2 )( T1Z + b2 ) = 0, ( q2T0Z−1 + c2 )( q2T0Z−1 + d2 ) = 0. Lemma 16.7 ([10, equations (3.5), (3.6)]). The elements T1, T0 of H̃ are invertible and T −11 = −a−2b−2T1 − 1− a−2b−2, T −10 = −q2c−2d−2T0 − 1− q2c−2d−2. From now on, assume P0 = ab+ a−1b−1, P1 = ab−1 + a−1b, (16.1) P2 = cd−1 + c−1d, P3 = q−1cd+ qc−1d−1. (16.2) Lemma 16.8. There exists an isomorphism of F-algebras Ĥq(P0, P1, P2, P3)→ H̃q(a, b, c, d) that sends t0 7→ −a−1b−1T1, t1 7→ −abT −11 Z−1, t2 7→ −q−2cdZT −10 , t3 7→ −qc−1d−1T0. The inverse isomorphism sends Z 7→ qt2t3, Z−1 7→ t0t1, T1 7→ −abt0, T0 7→ −q−1cdt3. Proof. One checks that the above maps are F-algebra homomorphisms, and that they are inverses. Consequently they are isomorphisms. � From now on, we identify the F-algebras Ĥq(P0, P1, P2, P3) and H̃q(a, b, c, d) via the isomor- phism in Lemma 16.8, and call the result H̃. In [10, equations (3.8), (3.9)] Koornwinder discusses two elements of H̃. The first is Y + q−2a2b2c2d2Y −1 where Y = T1T0. The second is Z + Z−1. These elements are related to A, B as follows. Lemma 16.9. In the algebra H̃, Z + Z−1 = A, Y + q−2a2b2c2d2Y −1 = q−1abcdB. Proof. Use Lemma 16.8. � In [10, Definition 6.1] Koornwinder defines an F-algebra ÃW (3, Q0) by generators and rela- tions. See also [11, Definition 2.5]. In [10, Corollary 6.3] Koornwinder displays an injection of F-algebras ÃW (3, Q0) → H̃. Consider the image of ÃW (3, Q0) under this injection. By con- struction and Lemma 16.9, this image is the subalgebra of H̃ generated by A, B, t0. In other words, the image is A. Thus [10, Definition 6.1, Corollary 6.3] yields a presentation of A by generators and relations, using the generators A, B, t0. The presentation looks as follows in terms of {Pi}i∈I. Theorem 16.10 ([10, Definition 6.1, Corollary 6.3]). The F-algebra A is presented by generators A, B, t0 and relations t0A = At0, t0B = Bt0, t20 = P0t0 − 1, The Universal Askey–Wilson Algebra and DAHA of Type (C∨1 , C1) 39 A2B − ( q2 + q−2 ) ABA+BA2 + ( q2 − q−2 )2 B + ( q − q−1 )2 Aγ = ( q − q−1 )( q2 − q−2 ) β, B2A− ( q2 + q−2 ) BAB +AB2 + ( q2 − q−2 )2 A+ ( q − q−1 )2 Bγ = ( q − q−1 )( q2 − q−2 ) α, ABAB (q2 − q−2)2 − BABA(q4 + 1 + q−4) (q2 − q−2)2 + B2A2 ( q2 + q−2 ) (q2 − q−2)2 +A2 ( q2 + q−2 ) +B2 ( q2 + q−2 ) + ABγ (q + q−1)2 + BAγ ( q−q−1 )( q3−q−3 ) (q2−q−2)2 − Aα ( q3−q−3 ) q2−q−2 − Bβ ( q3−q−3 ) q2−q−2 − γ2 (q + q−1)2 = ( q + q−1 )2 − (q−1t0 + qt−10 )2 − P 2 1 − P 2 2 − P 2 3 − ( q−1t0 + qt−10 ) P1P2P3, where α = ( q−1t0 + qt−10 ) P1 + P2P3, β = ( q−1t0 + qt−10 ) P3 + P1P2, γ = ( q−1t0 + qt−10 ) P2 + P3P1, t−10 = P0 − t0. Proof. Write [10, Definition 6.1] and [10, Corollary 6.3] in terms of A, B, t0 and {Pi}i∈I, using Lemma 16.9 together with (16.1), (16.2). � In this paper we presented several subalgebras of Ĥq and H̃ by generators and relations. We now compare these presentations. Theorems 4.1 and 4.5 together give a presentation of the subalgebra 〈A,B,C〉 of Ĥq by generators and relations, using the generators A, B, C. Theorem 14.1 gives a presentation of the subalgebra 〈A,B,C,T〉 of Ĥq by generators and rela- tions, using the generators A, B, C, t±10 , T1, T2, T3. Proposition 16.4 gives a presentation of the subalgebra A of H̃ by generators and relations, using the generators A, B, C, t0. Proposi- tion 16.5 and Theorem 16.10 each give a presentation of A by generators and relations, using the generators A, B, t0. We now discuss the logical implications between Proposition 16.4, Proposi- tion 16.5, and Theorem 16.10. Proposition 16.5 is discovered from Proposition 16.4 by partially eliminating C. Proposition 16.4 is discovered from Proposition 16.5 and the knowledge that C = C. Theorem 16.10 is discovered from Proposition 16.5 by eliminating C. Proposition 16.5 is readily verified using Theorem 16.10. However Proposition 16.5 is not readily discovered using Theorem 16.10 alone. Proposition 16.5 is discovered using Theorem 16.10 and the knowledge that C simplifies things. Neither C nor C appears in [10, 11]. In [11] Koornwinder discusses an algebra S(H̃) known as the spherical subalgebra of H̃. In [11, Theorem 3.2] Koornwinder displays an F-algebra isomorphism AW(3, Q0)→ S(H̃), where AW(3, Q0) is the homomorphic image of ÃW (3, Q0) described in [10, Section 2]. By [11, Sec- tion 3] the multiplicative identity of S(H̃) is a certain idempotent Psym in H̃. But Psym 6= 1, so S(H̃) and H̃ do not share the same 1. Therefore S(H̃) is not a subalgebra of H̃ according to our convention from Section 1. As far as we know, the results of the present paper are unrelated to S(H̃). 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Phys. 89 (1991), 1146–1157. http://dx.doi.org/10.1155/S1073792892000199 http://dx.doi.org/10.1090/conm/417/07922 http://arxiv.org/abs/math.QA/0304186 http://dx.doi.org/10.3842/SIGMA.2010.065 http://arxiv.org/abs/1001.2764 http://dx.doi.org/10.1016/j.jalgebra.2005.07.038 http://arxiv.org/abs/math.QA/0507477 http://dx.doi.org/10.1007/978-3-642-05014-5 http://dx.doi.org/10.3842/SIGMA.2007.063 http://arxiv.org/abs/math.QA/0612730 http://dx.doi.org/10.3842/SIGMA.2008.052 http://arxiv.org/abs/0711.2320 http://dx.doi.org/10.1155/S1073792804133072 http://arxiv.org/abs/math.RT/0306393 http://dx.doi.org/10.2307/121102 http://arxiv.org/abs/q-alg/9710032 http://dx.doi.org/10.3842/SIGMA.2011.069 http://arxiv.org/abs/1104.2813 http://dx.doi.org/10.3842/SIGMA.2011.099 http://arxiv.org/abs/1107.3544 http://dx.doi.org/10.1142/S0219498804000940 http://arxiv.org/abs/math.QA/0305356 http://dx.doi.org/10.1016/0550-3213(95)00361-U http://dx.doi.org/10.1016/0550-3213(95)00361-U http://arxiv.org/abs/cond-mat/9501129 http://dx.doi.org/10.1007/BF01015906 1 Introduction 2 The universal Askey-Wilson algebra 3 The universal DAHA q of type (C1,C1) 4 How q is related to q 5 Preliminaries concerning q 6 The proof of Theorems 4.1, 4.2, 4.3 7 A basis for the F-vector space q 8 The coefficient matrix 9 The proof of Theorem 4.4 10 Some results concerning algebraic independence 11 The structure of q 12 The proof of Theorem 4.5 13 The elements in q that commute with t0 14 A presentation for the algebra "426830A A,B,C,T"526930B 15 The center of q 16 Discussion References

The Universal Askey-Wilson Algebra and DAHA of Type (Cv₁,C₁)

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