The Askey-Wilson algebra and its avatars
Nicolas Crampé, Luc Frappat, Julien Gaboriaud, Loïc Poulain d'Andecy, Eric Ragoucy, Luc Vinet
TThe Askey–Wilson algebra and its avatars
Nicolas Cramp´e ∗ , Luc Frappat † , Julien Gaboriaud ‡ , Lo¨ıc Poulain d’Andecy § ,Eric Ragoucy ¶ , Luc Vinet (cid:107) , Institut Denis-Poisson CNRS/UMR 7013 - Universit´e de Tours - Universit´e d’Orl´eans,Parc de Grandmont, 37200 Tours, France. Laboratoire d’Annecy-le-Vieux de Physique Th´eorique LAPTh,Universit´e Grenoble Alpes, Universit´e Savoie Mont Blanc, CNRS, F-74000 Annecy, France. Centre de Recherches Math´ematiques, Universit´e de Montr´eal,P.O. Box 6128, Centre-ville Station, Montr´eal (Qu´ebec), H3C 3J7, Canada. Laboratoire de math´ematiques de Reims UMR 9008, Universit´e de Reims Champagne-Ardenne,Moulin de la Housse BP 1039, 51100 Reims, France.
October 1, 2020
Abstract:
The original Askey–Wilson algebra introduced by Zhedanov encodes the bispec-trality properties of the eponym polynomials. The name
Askey–Wilson algebra is currentlyused to refer to a variety of related structures that appear in a large number of contexts. Wereview these versions, sort them out and establish the relations between them. We focus ontwo specific avatars. The first is a quotient of the original Zhedanov algebra; it is shown to beinvariant under the Weyl group of type D and to have a reflection algebra presentation. Thesecond is a universal analogue of the first one; it is isomorphic to the Kauffman bracket skeinalgebra (KBSA) of the four-punctured sphere and to a subalgebra of the universal double affineHecke algebra ( C ∨ , C ). This second algebra emerges from the Racah problem of U q ( sl ) andis related via an injective homomorphism to the centralizer of U q ( sl ) in its threefold tensorproduct. How the Artin braid group acts on the incarnations of this second avatar throughconjugation by R -matrices (in the Racah problem) or half Dehn twists (in the diagrammaticKBSA picture) is also highlighted. Attempts at defining higher rank Askey–Wilson algebrasare briefly discussed and summarized in a diagrammatic fashion. Keywords:
Askey–Wilson algebra, Kauffman bracket skein algebra, U q ( sl ) algebra, doubleaffine Hecke algebra, centralizer, universal R -matrix, W ( D ) Weyl group, half Dehn twist. ∗ E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected] § E-mail: [email protected] ¶ E-mail: [email protected] (cid:107)
E-mail: [email protected] a r X i v : . [ m a t h . QA ] S e p ontents ( D ) symmetry 9 W ( D ) symmetry in the Racah algebra . . . . . . . . . . 10 U q ( sl ) and its centralizer in U q ( sl ) ⊗ U q ( sl ) and its universal R -matrix . . . . . . . . . . . . . . . . . . . . . . . . . 146.2 An algebra generated by the intermediate Casimir elements . . . . . . . . . . . 156.3 Fundamental theorems of invariant theory . . . . . . . . . . . . . . . . . . . . . 16 ( C ∨ , C )
168 Actions of the braid group 18 R -matrix and a braid group action on A . . . . . . . . 188.2 Half Dehn twists and the braid group action on Sk iq / (Σ , ) . . . . . . . . . . 198.3 Connection between both braid actions . . . . . . . . . . . . . . . . . . . . . . . 21 ( n ) algebra 22 aw ( n ) and Sk θ (Σ ,n +1 ) . . . . . . . . . . . . . . . . . . . . . . . . 24
10 Conclusion 25A Classical limit and injectivity 26
A.1 Polarised traces in U ( sl ) ⊗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27A.2 The algebra U α ( sl ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27A.3 Reduction to U ( sl ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27A.4 Racah algebra and diagonal centraliser in U ( sl ) ⊗ . . . . . . . . . . . . . . . . 282 Introduction
In order to provide an algebraic underpinning for the Askey–Wilson polynomials [1], Zhedanovintroduced what he called the Askey–Wilson algebra [2]. We shall refer to it rather as theZhedanov algebra. The Askey–Wilson polynomials sit at the top of the Askey classificationscheme of the hypergeometric orthogonal polynomials [3] and are, consequently, of fundamentalinterest; their algebraic interpretation by Zhedanov hence bears commensurate importance.These q -polynomials are bispectral: in addition to verifying a three-term recurrence prescribedby Favard’s theorem for any family of orthogonal polynomials [4], they are also eigenfunctions ofa q -difference operator. The Zhedanov algebra was constructed by taking these two bispectraloperators as generators and identifying the relations they obey. As sometimes happens withnatural constructs, related structures have emerged in a variety of contexts and have typicallyall been called Askey–Wilson algebras. This propensity keeps rising and it is hence timely toreview the topic. This paper will provide a taxonomy and a description of the algebras thatloosely go under the name of Askey–Wilson algebras and will characterize in some depth twoavatars of particular relevance. It will also set the stage for the exploration of generalizations.The focus of this survey will be on algebraic aspects. Before we discuss the contents inmore details, let us briefly go over some of the manifestations of these Askey–Wilson algebrasand the advances they have generated. Grosso modo, they have had direct applications inphysical models and have also been at the heart of mathematical developments establishinguseful interconnections between fields. One occurrence is in the recoupling of three irreduciblerepresentations of U q ( sl ) which is called its Racah problem. It is known that the 6 j -symbols ofthis algebra are expressed in terms of q -Racah polynomials which are a finite truncation of theAskey–Wilson ones. As a rule, whenever the Askey–Wilson polynomials (or their truncatedversion) appear, the associated algebra will be present. In the case of the Racah problem, itis found that the intermediate Casimir elements verify Askey–Wilson relations [5, 6]. Thesepolynomials and algebras appear in the study of the ASEP model with open boundaries [7], asmartingale polynomials and quadratic harnesses in probabilistic models [8] and are connectedto (a degeneration of) the Sklyanin algebra [9–11]. Quite generally, the Askey–Wilson alge-bras are present in the context of integrable models, through the Yang–Baxter and reflectionequations [12–17], and can be viewed as truncations of the q -Onsager algebra [12]. Elementsof representation theory have been investigated in [2, 6, 18–20] and another of its manifes-tations is as a coideal subalgebra of U q ( sl ) [21–23]. The Askey–Wilson algebras have alsobeen cast in the framework of Howe duality using the pair ( U q ( sl ) , o q / (2 n )) [24–27]; theyare special cases of the recently introduced Painlev´e algebras [28] and belong to the Calabi–Yau class [29]. There is a significant connection to the field of algebraic combinatorics, asAskey–Wilson algebras are central in the classification of P - and Q - polynomial associationschemes and the study of Leonard pairs and triples [30–35]. The Askey–Wilson algebras havealso been shown to offer a promising platform to extend the quantum Schur–Weyl duality toarbitrary representations and have been seen in that respect to admit the Temperley–Lieb andBirman–Murakami–Wenzl algebras [36] as quotients. Askey–Wilson algebras have moreoverfound their way in the general framework of knot theory through their identification with theKauffman bracket skein algebras of the four-punctured sphere Sk iq / (Σ , ) and other elemen-tary surfaces [37–39]. This is also closely connected to double affine Hecke algebras (DAHA) asthe Askey–Wilson algebra is related to the spherical subalgebra of the DAHA of type ( C ∨ , C )[20, 28, 40–46].This overview of the relevance of Askey–Wilson algebras in different domains motivates thepresent topical report. Let us make at this point a few additional remarks on the introduction3f the algebra Sk iq / (Σ , ) in the Askey–Wilson picture to stress that this paper also featuresnovel results relating the Askey–Wilson algebra, the Kauffman bracket skein algebra and thebraid group.Kauffman bracket skein algebras (KBSA) have been defined independently by Turaev [47]and Bullock and Przytycki [37] in the study of knot invariants and can be seen to encompassthe celebrated Jones polynomial [48, 49]. Computations in the KBSA are done through dia-grammatic manipulations given by a set of rules (the skein relations). It is appreciated thatthis Sk iq / (Σ , ) algebra is closely related to the centralizer of U q ( sl ) in its threefold tensorproduct. This ties in with the Temperley–Lieb algebra which admits a diagrammatic presenta-tion [49–51] for generic q , is precisely the centralizer of U q ( sl ) in the threefold tensor productof the fundamental representations of U q ( sl ) [52] and, as already indicated, was found to bea quotient of the Askey–Wilson algebra [36].A natural question that has arisen asks about higher rank extensions of Askey–Wilsonalgebras. In view of the ubiquity of the 3-generated Askey–Wilson algebras it is to be ex-pected that such generalizations will prove quite fruitful. This question is non-trivial howeversince many avenues that are likely to yield different outcomes can be followed. Among thosepossibilities, one is to consider the algebra realized by the intermediate Casimir elements inmultifold tensor products of U q ( sl ) [53–56], and another is to increase the rank of the algebra U q ( sl ) to, say, U q ( sl ) when studying the Racah problem. Augmenting the number of punc-tures of the sphere in the KBSA approach could also be envisaged. Making much sense is theidea to start from the multivariate Askey–Wilson polynomials [57], to work out the algebraformed by its bispectral operators [55, 58, 59] and to take things from there. This is after allhow the story began. Steps have been taken in these directions but final conclusions have notbeen reached. Some authors have considered higher order truncations of the reflection algebra[60] understood as a quotient of the q -Onsager algebra (see also [61] for the classical limit ofthis result). The upshot is that there is currently no clear consensus on what the higher rankAskey–Wilson algebra is ∗ . This is not too surprising since there are still a few loose ends inthe rank one cases.As a prelude to a solid understanding of the higher rank Askey–Wilson algebra, it isappropriate to clarify the picture for the ordinary Askey–Wilson algebras. Indeed, as thesealgebras have appeared in multiple instances in the literature, names, conventions and notationsare quite diverse. We are here proposing a standardization and offering a number of new results.The paper will unfold as follows. The various Askey–Wilson avatars will be introduced inSection 2. They will be given names and defined in a comparative way. Emphasis will beput on two particular versions. The first is a quotient of the Zhedanov algebra which we willcall the Special Zhedanov algebra. In Section 3, we will show that the Zhedanov algebra isobtained as the reflection algebra defined from particular R - and reflection matrices. In thisformalism, the Special Zhedanov algebra corresponds to fixing the Sklyanin determinant toa certain value; the name Special is chosen in analogy with the nomenclature of Lie groups.A Weyl group W ( D ) symmetry of the Special Zhedanov algebra will then be presented inSection 4, thus generalizing an analoguous result for the Racah algebra. The second avatarthat will be closely looked at will be called the Special Askey–Wilson algebra. It can beseen as the equivalent of the Special Zhedanov algebra where the parameters are promotedto central elements in the algebra. That this algebra is isomorphic to the Kauffman bracketskein algebra of the four-punctured sphere Sk iq / (Σ , ) is the object of Section 5. In Section ∗ Remarkably, for the q → q → −
4, the Special Askey–Wilson algebra will further be related to the algebra A associated to theRacah problem of U q ( sl ) and to the centralizer C of U q ( sl ) in its threefold tensor product.An injective homomorphism of algebras between the latter two structures will be stated andits proof will be found in Appendix A. The relation between the Special Askey–Wilson algebraand the universal double affine Hecke algebra (DAHA) of type ( C ∨ , C ) will be discussed inSection 7. How the Artin braid group B acts on both the A and Sk iq / (Σ , ) algebras,respectively through conjugation by braided R -matrices and through half Dehn twists willbe highlighted in Section 8. The question of the possible higher-rank generalizations of theAskey–Wilson algebra will be addressed in Section 9. A crossing index will be introduced andused to summarize efficiently the main results of [53] and [56] and new relations for the higherrank analogues will be provided. Elements of interest for further study of the higher rankgeneralizations of the Special Askey–Wilson algebra will be offered in addition. Concludingremarks will end the paper. As mentioned in the above, the name
Askey–Wilson algebra has appeared and been connectedto diverse objects in a multitude of contexts. Therefore, the notations and appellations inthe literature are sometimes confusing. For the sake of clarity, we start by presenting thesedifferent algebraic structures and give to them unambiguous names to distinguish them.
The Askey–Wilson algebra aw (3) is the unital associative algebra depending on theparameter q with generators C , C , C and central elements C , C , C , C obeying the Z -symmetric relations C + [ C , C ] q q − q − = C C + C C q + q − , (2.1a) C + [ C , C ] q q − q − = C C + C C q + q − , (2.1b) C + [ C , C ] q q − q − = C C + C C q + q − , (2.1c)where the q -commutator is defined by [ A, B ] q = qAB − q − BA . Throughout the paper, wesuppose that q ∈ C is not a root of unity. The Casimir element of this algebra is Ω := qC C C + q C + q − C + q C − qC ( C C + C C ) − q − C ( C C + C C ) − qC ( C C + C C ) . (2.1d)From this algebra, we define multiple quotients or subalgebras which appear in different con-texts: The Special Askey–Wilson algebra saw (3) is the quotient of aw (3) by the supplementaryrelation Ω = ( q + q − ) − C − C − C − C − C C C C . (2.2)A justification of the adjective special is given in Section 3. This algebra is isomorphic to theKauffman bracket skein module of the four-punctured sphere (see Section 5) and is directly5ssociated to the centralizer of the diagonal action of U q ( sl ) in its threefold tensor product(see Section 6). The universal Askey–Wilson algebra ∆ q defined in [33] is the subalgebra of aw (3)generated by C , C , C as well as the central elements α = C C + C C , β = C C + C C and γ = C C + C C . The Casimir element of ∆ q becomesΩ = qC C C + q C + q − C + q C − qC α − q − C β − qC γ. (2.3)Its finite irreducible representations have been classified in [19]. The evaluated Askey–Wilson algebra Z q ( m , m , m ) is the quotient of aw (3) by thesupplementary relations C i = q m i + q − m i , i = 1 , , . (2.4)It plays a central role in the study of the centralizer of the diagonal embedding of U q ( sl ) inthe threefold tensor product of representations of U q ( sl ) [36]. The Zhedanov algebra Zh q ( m , m , m , m ) is the quotient of aw (3) by C i = q m i + q − m i , C = q m + q − m , i = 1 , , , (2.5)and was first introduced by Zhedanov as the algebra encoding the bispectrality of the Askey–Wilson polynomials [2]. An alternative presentation of its defining relations is given in (4.10a)–(4.10c), (4.10e)–(4.10g). The Special Zhedanov algebra sZh q ( m , m , m , m ) is obtained as the quotient of saw (3) by relations (2.5) (see (4.10a)–(4.10h) for an alternative presentation). It appearsnaturally as the commutation relations of the intermediate Casimir elements acting on themultiplicity space of the decomposition of the threefold tensor product of representations of U q ( sl ) (see Section 6.3). PBW basis
The Askey–Wilson algebra aw (3) has a Poincar´e–Birkhoff–Witt (PBW) basis given explicitlyby the following elements C i C j C k C m C n C p C q , i, j, k, m, n, p, q ∈ N . (2.6)We can also obtain a PBW basis for the Special Askey–Wilson algebra saw (3) from the oneof aw (3) by restricting the range of the exponent j to { , } instead of N . The PBW basis foruniversal Askey–Wilson algebra ∆ q has been given in [33].6 alabi–Yau algebra The Zhedanov algebra Zh q ( m , m , m , m ) can be derived from a Calabi–Yau potential in thefollowing sense [64]. Let F = C [ x , x , x ] be a free associative algebra and view F as a gradedalgebra such that deg ( x ) = d , deg ( x ) = d and deg ( x ) = d (with 0 < d ≤ d ≤ d ). Wedefine F cycl = F/ [ F, F ] and the map ∂∂x j : F cycl → F on cyclic words as follows ∂ [ x i x i . . . x i r ] ∂x j = (cid:88) { s | i s = j } x i s +1 x i s +2 . . . x i r x i x i . . . x i s − (2.7)and we extend it to F cycl by linearity. Let Φ( x , x , x ) ∈ F cycl be a potential which can bedecomposed as follows Φ( x , x , x ) = Φ ( d ) ( x , x , x ) + Φ 00 0 0 uq − uq . (3.1)7his R -matrix is associated to the quantum affine algebra U q ( (cid:98) sl ) and is a solution of theYang–Baxter equation R ( u /u ) R ( u /u ) R ( u /u ) = R ( u /u ) R ( u /u ) R ( u /u ) , (3.2)where R = (cid:80) a R a ⊗ R a ⊗ , R = (cid:80) a ⊗ R a ⊗ R a , R = (cid:80) a R a ⊗ ⊗ R a if one writes R = (cid:80) a R a ⊗ R a and as the 2 × B ( u ) = uqC − C uq + p /u + p (cid:48) uu − /u qu + 1 qu − [ C , C ] q q − /q + p (cid:48)(cid:48) q + 1 /q − qu − qu + [ C , C ] q q − /q − p (cid:48)(cid:48) q + 1 /q uqC − C uq + p u + p (cid:48) /uu − /u , (3.3)where we refer to (4.7a)–(4.7c) for the definition of p , p (cid:48) and p (cid:48)(cid:48) . Proposition 3.1. [12] The set of relations obtained from the reflection equation R ( u/v ) B ( u ) R ( uv ) B ( v ) = B ( v ) R ( uv ) B ( u ) R ( u/v ) , (3.4) where B ( u ) = B ( u ) ⊗ and B ( u ) = ⊗ B ( u ) , is equivalent to the defining relations of Zh q ( m , m , m , m ) .Proof. We look at each matrix element of the reflection equation (3.4) and derive 16 relations.For each or them, we extract the different coefficients w.r.t. the parameter u ; this providesrelations between C and C . By direct investigation, we verify that all the obtained relationsare equivalent to the defining relations of Zh q ( m , m , m , m ).Rephrasing this proposition, the Zhedanov algebra Zh q ( m , m , m , m ) is isomorphic tothe truncated reflection algebra defined by the R -matrix (3.1) and the truncated reflectionmatrix (3.3). Remark 3.2. There exists a more general form for the reflection matrix, containing an infinitenumber of generators encompassed in formal series of u and u . The elements of the reflectionmatrix (3.3) can be obtained as a truncation of these formal series. The algebra defined by thegeneral reflection matrix obeying the reflection equation (3.4) is isomorphic to the q -Onsageralgebra [14]. Therefore, the Zhedanov algebra can also be seen as a quotient of the q -Onsageralgebra. In the context of the reflection algebra it is well-known how to obtain central elements [65].Indeed, let us define the Sklyanin determinant sdet B ( u ) as followssdet B ( u ) := − tr (cid:0) R (1 /q ) B ( u/q ) R ( u /q ) B ( u ) (cid:1) . (3.5)We can show that the coefficients of sdet B ( u ) commute with C and C . We recover in thisway that the operator Ω given by (2.1d) commutes with C and C . The Sklyanin determi-nant gives solely Ω as a central element. Fixing the Sklyanin determinant to an appropriatevalue allows us to give a FRT presentation of sZh q ( m , m , m , m ):8 roposition 3.3. The truncated reflection algebra defined by the R -matrix (3.1) , the truncatedreflection matrix (3.3) and quotiented by the relation sdet B ( u ) = q (1 − q ) ( u + q − m − m )( u + q m + m )( u + q m − m )( u + q m − m ) × ( u + q − m − m )( u + q m + m )( u + q m − m )( u + q m − m ) , (3.6) is isomorphic to sZh q ( m , m , m , m ) .Proof. By direct computations, we show that (3.6) is equivalent to imposing (2.2).The fact that sZh q ( m , m , m , m ) can be defined as a truncated reflection algebra wasexpected, but it is a surprise that the r.h.s. of (3.6) factorizes into such a simple form. ( D ) symmetry The algebra sZh q ( m , m , m , m ) has a remarkable symmetry based on the Weyl group W ( D ) associated to the Lie algebra D . To describe it, let us introduce a root system oftype D and fix a set of simple roots α , α , α , α with labeling according to the followingDynkin diagram: 1 3 42The Weyl group W ( D ) is generated by the reflections s i associated to the simple roots α i which satisfy, for 1 ≤ i, j ≤ s i =1 s i s j = s j s i s i s j s i = s j s i s j if i and j are not connected in the Dynkin diagram , if i and j are connected in the Dynkin diagram . (4.1)Its order is 192. Let us now associate the parameters m , m , m , m with some of the rootsas follows: m = α , m = α , m = α , m = Θ , (4.2)where Θ is the longest positive root. The explicit expression of Θ is: α = 12 ( m − m − m − m ) . (4.3)It is elementary to calculate the actions s i expressed in terms of the parameters: s : m (cid:55)→ − m , s : m (cid:55)→ − m , s : m (cid:55)→ − m , s : m (cid:55)→ m + α ,m (cid:55)→ m + α ,m (cid:55)→ m + α ,m (cid:55)→ m − α , (4.4)where the omitted actions are trivial and the explicit expression of α is given above. Theaction of the Weyl group is extended to any function as follows:( σf )( m , m , m , m ) = f ( σ ( m ) , σ ( m ) , σ ( m ) , σ ( m )) (4.5)for σ ∈ W ( D ). 9 roposition 4.1. The Weyl group W ( D ) is a symmetry of sZh q ( m , m , m , m ) i.e. sZh q ( m , m , m , m ) = sZh q ( σ ( m ) , σ ( m ) , σ ( m ) , σ ( m )) , (4.6) for any σ ∈ W ( D ) .Proof. In sZh q ( m , m , m , m ), we remark that the only functions of m i which appear are p = χ m χ m + χ m χ m , (4.7a) p (cid:48) = χ m χ m + χ m χ m , (4.7b) p (cid:48)(cid:48) = χ m χ m + χ m χ m , (4.7c) p = χ m + χ m + χ m + χ m + χ m χ m χ m χ m , (4.7d)where χ m = q m + q − m . By direct computations, we can show that these functions are invariantby the transformations s , s , s and s given by (4.4), which concludes the proof since theygenerate W ( D ).In the study of the finite representations of the universal algebra ∆ q a W ( D ) symmetryhas been also investigated [67, 68]. W ( D ) symmetry in the Racah algebra Let us perform the transformation K I = C I − ( q + q − )( q − q − ) , (4.8)with I ∈ { , , , , , } . Note that 13 does not belong to this set. In the algebra sZh q ( m , m , m , m ), one gets, for i = 1 , , , K i = χ m i − ( q + q − )( q − q − ) = (cid:104) m i (cid:105) q − (cid:20) (cid:21) q , (4.9)where the q -number is defined by [ m ] q = q m − q − m q − q − . The commutation relations of the algebra sZh q ( m , m , m , m ) become[ K , K ] q = K , (4.10a)[ K , K ] q = ( q + q − ) (cid:0) − { K , K } − K + ξ K + ξ (cid:1) , (4.10b)[ K , K ] q = ( q + q − ) (cid:0) − { K , K } − K + ξ K + ξ (cid:48) (cid:1) , (4.10c)and the supplementary relation becomes − q q − q − q + q − K K K − qK K K − q − K K K + q ( q + q − ) K + (cid:16) ξ q + q − − (cid:17) { K , K } + qξ K + q − ξ (cid:48) K = ξ − ξ − ξ (cid:48) − ξ , (4.10d)10ith ξ = 1 q + q − (cid:16) M + M + M + M − 1) + ( q − q − ) ( M M + M M ) (cid:17) , (4.10e) ξ = ( M − M )( M − M ) , (4.10f) ξ (cid:48) = ( M − M )( M − M ) , (4.10g) ξ = ( M M − M M )( M − M + M − M ) + ( q − q − ) ( M M − M M ) , (4.10h)where we use the notation M i = (cid:2) m i (cid:3) q . As expected, we can check that the functions ξ , ξ , ξ (cid:48) and ξ are invariant under the action of the Weyl group W ( D ).The advantage of this presentation of sZh q ( m , m , m , m ) is that the classical limit q → W ( D ) action also holds for theRacah algebra and we recover the results of [69]. Remark 4.2. In the classical limit q → , the functions ξ , ξ , ξ (cid:48) and ξ form a basis forpolynomials invariant under the action of W ( D ) , as expected. In the generic case ( q ∈ C , nota root of unity), one gets two different sets of invariant functions: S ξ = { ξ , ξ , ξ (cid:48) , ξ } on theone hand and S p = { p , p (cid:48) , p (cid:48)(cid:48) , p } on the other hand. We have checked that there exists aninvertible polynomial mapping between these two sets. However, only S ξ admits a non-trivialclassical limit. Kauffman bracket skein module quantizations have been introduced in [37, 47] and furtherstudied along our lines of interest for this paper in [39, 70, 71]. We will now recall some keydefinitions and results from these investigations. We shall work with an oriented 3-manifold M which is a thickened surface, that is M = Σ ,n × I , where I = [0 , 1] and Σ ,n is the n -puncturedsphere. Definition 5.1. The quantized skein module Sk θ ( M ) is the C [ θ ± ] -module spanned by framedand unoriented links in M modulo the Kauffman bracket skein relations that allow to “simplifythe crossings”: = θ + θ − , (5.1a)= − ( θ + θ − ) , (5.1b) where θ ∈ C is not a root of unity and in the framing relation (5.1b) the link should not enclosea puncture. This defines an algebra, which we will denote Sk θ (Σ ,n ) , for which multiplicationis given by stacking the links on top of each other in the I direction. We shall use diagrams that correspond to the projection of the links on the surface (all thewhile keeping the information about the relative “height” of the links in the I direction). Letus now establish the conventions for these drawings (framed links diagrams).11he n -punctured sphere Σ ,n is equivalent to the plane with n − n − 1) drawn × ’s): × × × . . . × (5.2)The dashed contour corresponds to the n th puncture of the sphere. We will omit the contourin the subsequent diagrams but it is always understood to be there.Framed links that enclose punctures are represented by loops drawn around the × ’s. Weshall use the term “loops” to refer unambiguously to the framed links in the remainder of thepaper. These loops can be homotopically deformed without crossing the holes (punctures).Remark that loops enclosing a single puncture are central elements in Sk θ (Σ ,n ). This is alsotrue for the n th puncture, which amounts to saying that the loop enclosing the ( n − 1) punctures × is also central.Let us now consider the surface Σ , and give names to a few loops: × × × = A × × × = A × × × = A × × × = A × × × = A × × × = A × × × = A (5.3)Following the definition, multiplication of two loops X · Y means putting Y on top of X , forexample: × × × A · A = (5.4)One would then proceed to use relations (5.1) to simplify the expressions: A · A = θ (cid:16) × × × (cid:17) − θ − (cid:16) × × × (cid:17) = θ × × × + (cid:16) × × × (cid:17) + (cid:16) × × × (cid:17) + θ − × × × . = θ A + A · A + A · A + θ − × × × . (5.5)Similarly, exchanging the order of multiplication, one obtains the same diagrams but withinverse coefficients: A · A = θ − A + A · A + A · A + θ × × × . (5.6)12e see immediately that one gets θ A · A − θ − A · A = ( θ − θ − ) A + ( θ − θ − )( A · A + A · A ) . (5.7)The skein algebra Sk θ (Σ , ) is directly linked to the Askey–Wilson algebra as stated in thefollowing proposition: Proposition 5.2. The Special Askey–Wilson algebra saw (3) is isomorphic to the Kauffmanbracket skein algebra Sk iq / (Σ , ) . The isomorphism is given by the following invertible map: C I (cid:55)→ A I , (5.8) for I ∈ { , , , , , , } .Proof. The isomorphism is directly verified by comparing the relations of saw (3) and the onesof the Kauffman bracket skein algebra obtained in [37] (see also Proposition 3 . saw (3).Let us emphasize that the previous isomorphism involves the Special Askey–Wilson algebra saw (3). If we replace saw (3) by aw (3) in the map of the proposition, the homomorphismwould be not injective and if we instead replace saw (3) by ∆ q (as in [6, 72]), it would be notsurjective.One notes that the Z -symmetry of the saw (3) relations is made manifest in terms of theframed links picture, as the punctures do not have fixed positions and can be switched around.From now on we will unambiguously refer to the drawn loops identified as the generatorsof Sk iq / (Σ , ) directly as their C I counterpart following (5.8). This correspondence (5.8)leads to a natural labeling of the punctures. Indeed, consider the generators given in (5.3):the punctures enclosed in a given loop correspond precisely to the set of indices I of thecorresponding generator C I if one labels the punctures consecutively as: × × × (5.9) Remark 5.3. We recall that one arrives to the Special Zhedanov algebra sZh q ( m , m , m , m ) from the Special Askey–Wilson algebra saw (3) by attributing a value to the central elements C i , i = 1 , , , , see (2.5) . In the same way, starting from the Kauffman bracket skeinalgebra Sk iq / (Σ , ) , one can define an evaluated Kauffman bracket skein algebra, denoted Sk iq / (Σ , ; m , m , m , m ) by attributing a value to the puncture-framing relations: × i = q m i + q − m i , i = 1 , , , × × × = q m + q − m . (5.10) Note that the last drawing corresponds in fact to a contour enclosing the fourth puncture on thesphere, see (5.2) . As a corollary of Proposition 5.2, the algebra Sk iq / (Σ , ; m , m , m , m ) is isomorphic to the Special Zhedanov algebra sZh q ( m , m , m , m ) .Relations (5.10) with m i = 1 already appear in the definition of the skein algebra of arcsand link introduced in [73], from where we borrowed the terminology ‘puncture-framing’. U q ( sl ) and its centralizer in U q ( sl ) ⊗ The goal of this section is to discuss the notion of centralizer of U q ( sl ) in U q ( sl ) ⊗ , which wedenote by C , and connect it with the Special Askey–Wilson algebra saw (3). U q ( sl ) and its universal R -matrix Let us fix the notation and conventions that will be used to perform the explicit calculationsin U q ( sl ) (note that the results obtained will be independent of these conventions at the end).We shall first define the quasi-triangular Hopf algebra U q ( sl ), present its braided universal R -matrix and list some additional properties of interest. U q ( sl ) is an associative algebra generated by E , F , q H and q − H obeying the definingrelations q H q − H = q − H q H = 1 , q H E = qEq H , q H F = q − F q H and [ E, F ] = [2 H ] q . (6.1)The center of this algebra is generated by the following Casimir element (denoted Λ in [54,56]) Q = ( q − q − ) (cid:18) F E + qq H + q − q − H ( q − q − ) (cid:19) . (6.2)The algebra U q ( sl ) can be endowed with a Hopf structure. In particular, its comultiplication(or coproduct) homomorphism ∆ : U q ( sl ) → U q ( sl ) ⊗ U q ( sl ) is given by∆( E ) = E ⊗ q − H + q H ⊗ E, ∆( q H ) = q H ⊗ q H , (6.3a)∆( F ) = F ⊗ q − H + q H ⊗ F, ∆( q − H ) = q − H ⊗ q − H , (6.3b)and is coassociative (∆ ⊗ id)∆ = (id ⊗ ∆)∆ . (6.4)The quantum algebra U q ( sl ) is called quasi-triangular because in a completion of U q ( sl ) ⊗ U q ( sl ), there exists a universal R -matrix R which is invertible and satisfies∆( x ) R = R ∆ op ( x ) for x ∈ U q ( sl ) , (6.5)(id ⊗ ∆) R = R R , (6.6)(∆ ⊗ id) R = R R , (6.7)where in the Sweedler notation we write the opposite comultiplication ∆ op ( x ) = x (2) ⊗ x (1) if∆( x ) = x (1) ⊗ x (2) . In the previous relation, we have used the notations R = R α ⊗ R α ⊗ R = 1 ⊗ R α ⊗ R α and R = R α ⊗ ⊗ R α where R = R α ⊗ R α (the sum over repeatedindices α is understood). The universal R -matrix is given explicitly by [74] R = q H ⊗ H ) ∞ (cid:88) n =0 ( q − q − ) n [ n ] q ! q n ( n − / (cid:0) Eq H ⊗ q − H F (cid:1) n , (6.8)where [ n ] q ! = [ n ] q [ n − q . . . [2] q [1] q and, by convention, [0] q ! = 1.One can also define the so-called braided universal R -matrix ˇ R byˇ R i = R i,i +1 σ i,i +1 (6.9)14here σ i,i +1 acts on the i th and ( i + 1) th factors of the tensor product as σ i,i +1 ( · · · ⊗ x i ⊗ x i +1 ⊗ . . . ) = ( · · · ⊗ x i +1 ⊗ x i ⊗ . . . ) σ i,i +1 . (6.10)This braided universal R -matrix satisfies the braided Yang–Baxter equationˇ R i ˇ R i +1 ˇ R i = ˇ R i +1 ˇ R i ˇ R i +1 . (6.11) Let us define the following intermediate Casimir elements Q = Q ⊗ ⊗ , Q = 1 ⊗ Q ⊗ , Q = 1 ⊗ ⊗ Q,Q = ∆( Q ) ⊗ Q (1) ⊗ Q (2) ⊗ , Q = 1 ⊗ ∆( Q ) = 1 ⊗ Q (1) ⊗ Q (2) ,Q = (∆ ⊗ id)∆( Q ) . (6.12)The labeling of these intermediate Casimir elements is chosen so as to refer to the non-trivialfactors in the tensor product U q ( sl ) ⊗ . Definition 6.1. The algebra A is the subalgebra of U q ( sl ) ⊗ generated by the intermediateCasimir elements Q , Q , Q , Q , Q and Q . Let us define an additional intermediate Casimir element Q = ˇ R − Q ˇ R = ˇ R Q ˇ R − . (6.13)It has been proven in [75] that this element is in U q ( sl ) ⊗ (and not in its completion), thatthe second equality is compatible with the first one and that the following proposition holds: Proposition 6.2. The intermediate Casimir elements Q , Q , Q , Q , Q , Q and Q belong to the centralizer C of the diagonal action of U q ( sl ) in U q ( sl ) ⊗ defined by C = { X ∈ U q ( sl ) ⊗ (cid:12)(cid:12) [(∆ ⊗ id)∆( x ) , X ] = 0 , ∀ x ∈ U q ( sl ) } . (6.14)The precise links between the Askey–Wilson algebra, the centralizer and the algebra A generated by the intermediate Casimir elements are given in the following proposition. Proposition 6.3. The algebra saw (3) has an homomorphic injective image in C . The map-ping is done as follows: C I (cid:55)→ Q I , for I ∈ { , , , , , , } . (6.15) The algebra saw (3) is isomorphic to A .Proof. All the relations of saw (3) given by (2.1) and (2.2) are easily checked in U q ( sl ) ⊗ upon rewriting the Q I ’s in terms of the U q ( sl ) ⊗ generators. The proof of the injectivity ispostponed to Appendix A. Since the algebra A is the image of the map (6.15), it follows that saw (3) is isomorphic to A .This realization of the Askey–Wilson algebra in U q ( sl ) ⊗ was the motivation for adding therelation (2.2) to the “intuitive” set of relations of aw (3). Indeed, since relation (2.2) is obeyedby the intermediate Casimir elements, it should also be included in the algebra encoding theproperties of these Casimir elements. 15 orollary 6.4. The algebra A is isomorphic to the Kauffman bracket skein algebra of thefour-punctured sphere Sk iq / (Σ , ) . The isomorphism is given by the following map: φ : Q I (cid:55)→ A I , for I ∈ { , , , , , , } . (6.16) Proof. A direct consequence of the Propositions 5.2 and 6.3. In the previous section, we introduced the centralizer C of the diagonal action of U q ( sl ) in thethreefold tensor product and showed its connection with the Askey–Wilson algebra saw (3).We now focus on similar objects in the case where we represent each factor U q ( sl ) in U q ( sl ) ⊗ by a finite-dimensional irreducible representation.The quantum algebra U q ( sl ) has finite irreducible representations of dimension m = 2 j + 1that we will denote by M ( m ), with m ∈ Z > . The name “spin- j representation” is usuallyused to refer to M ( m = 2 j + 1). The representation map will be denoted by π m : U q ( sl ) → End( M ( m )). The representation of the Casimir element (6.2) in the space M ( m ) is π m ( Q ) = χ m m , (6.17)where χ m = q m + q − m and m is the m × m identity matrix.From now on, we fix three integers m , m and m . The threefold tensor product ofirreducible representations of U q ( sl ) decomposes into the following direct sum of irreduciblerepresentations M ( m ) ⊗ M ( m ) ⊗ M ( m ) = (cid:77) m M ( m ) ⊗ V m m ,m ,m , (6.18)where V m m ,m ,m is called the multiplicity space. We recall that we look at cases where q is nota root of unity otherwise the previous statement would be wrong.We now fix four integers m , m , m , m and denote by Q I the image of Q I in V m m ,m ,m (for I ∈ { , , , , , , } ). We get Q = χ m , Q = χ m , Q = χ m and Q = χ m . Proposition 6.5. There exists a surjective algebra homomorphism from sZh q ( m , m , m , m ) to End ( V m m ,m ,m ) given by C I (cid:55)→ Q I , for I ∈ { , , } . (6.19)This proposition which provides the generators for the centralizer of the diagonal action issometimes called in invariant theory the “first fundamental theorem”. The map in the previousproposition is not injective. The description of the kernel of this map is the subject of [36] (seealso [76]) and is called the “second fundamental theorem”.We recall that the algebra sZh q ( m , m , m , m ) possesses a W ( D )-symmetry. Let usremark that a similar Weyl group symmetry of type E has been discovered recently [77] inthe case of the centralizer of the diagonal embedding of U ( sl ) in two copies of U ( sl ). ( C ∨ , C ) Double affine Hecke algebras (DAHA) of type ( C ∨ , C ) were introduced in [78] and theirconnections with Askey–Wilson polynomials were first explored in [18] and [40]. Universalanalogues of these DAHA were later introduced and studied in [20, 43, 44].In this section, we present another connection between the Special Askey–Wilson algebra saw (3) and a certain subalgebra of a universal DAHA of type ( C ∨ , C ).16 efinition 7.1. We introduce the following algebras • The universal Double Affine Hecke Algebra of type ( C ∨ , C ) [43] is defined as the asso-ciative algebra (cid:98) H q with generators { t ± i , i = 0 , . . . , } and relations: t i t − i = t − i t i = 1 , (7.1a) t i + t − i is central , (7.1b) t t t t = q − . (7.1c) The “usual” DAHA, denoted H q ( k , k , k , k ) , is recovered when the central elements t i + t − i have complex values k i + k − i , with k i (cid:54) = 0 . • The algebra Γ q [44] is the subalgebra of (cid:98) H q commuting with the distinguished generator t ( Γ q is the centralizer of t in (cid:98) H q ): Γ q = { h ∈ (cid:98) H q | [ h, t ] = 0 } . (7.2) • Let e be the following idempotent of H q ( k , k , k , k ) [71] e = t − k k − − k . (7.3) The spherical DAHA, denoted SH q ( k , k , k , k ) [41, 42], is defined as SH q ( k , k , k , k ) = e H q ( k , k , k , k ) e . (7.4)The following theorems relate DAHA to the previously introduced algebraic structures. Theorem 7.2. [44] The map Θ : saw (3) → Γ q defined by C (cid:55)→ t t + ( t t ) − ,C (cid:55)→ t t + ( t t ) − ,C (cid:55)→ t t + ( t t ) − , C (cid:55)→ t + t − ,C (cid:55)→ t + t − ,C (cid:55)→ t + t − ,C (cid:55)→ q − t + qt − . (7.5) is an injective algebra homomorphism. Theorem 7.3. (Theorem 3.2 in [42]) The Special Zhedanov algebra sZh q ( m , m , m , m ) isisomorphic to the spherical DAHA SH q ( k , k , k , k ) . Remark 7.4. Spherical DAHAs have also been connected to skein algebras of higher genus.The Kauffman bracket skein algebra of the once-punctured torus Sk θ (Σ , ) is related to a (spher-ical) DAHA of type A [37, 79] and the genus two skein algebra is related to a genus twospherical double affine Hecke algebra in [80]. Actions of the braid group In this section, we provide two actions of the braid group: the first one on the algebra A and the second one on the skein algebra Sk iq / (Σ , ). Then, we show how these two actionsare compatible and give a diagrammatic presentation of the intermediate Casimir elements of U q ( sl ) ⊗ .We recall that the braid group on n strands B n is generated by the elements s , . . . , s n − as well as their inverses s − , . . . , s − n − satisfying s i s i +1 s i = s i +1 s i s i +1 ,s i s j = s j s i if | i − j | ≥ ,s − i s i = s i s − i = 1 . (8.1) R -matrix and a braid group action on A Let us recall that we define the generators Q as follows Q = Q d = ˇ R − Q ˇ R = ˇ R Q ˇ R − . (8.2)From the result of Proposition 6.3, we know that Q satisfies Q = Q Q + Q Q q + q − − [ Q , Q ] q q − q − , (8.3)and is in the algebra A which is generated by Q , Q , Q , Q , Q and Q . Now from (8.2)it is natural to consider the following element which is analogous to Q d : Q u = ˇ R Q ˇ R − = ˇ R − Q ˇ R . (8.4)It has been shown in [75] that this element is also in A since it can be obtained as Q u = Q Q + Q Q q + q − − [ Q , Q ] q q − q − . (8.5)The labels u and d added on the Casimir elements Q d and Q u stand for up and down .These names come from the form of their image in Sk iq / (Σ , ) given in Corollary 6.4: × × × = φ ( Q u ) , (8.6a) × × × = φ ( Q d ) . (8.6b)This procedure of obtaining additional elements of A by conjugations of braided R -matricescan be described by an automorphism action. Let us define the following automorphisms of A denoted Ψ s i and Ψ s − i byΨ s i ( X ) = ˇ R i X ˇ R − i and Ψ s − i ( X ) = ˇ R − i X ˇ R i = Ψ − s i ( X ) , (8.7)18or i = 1 , X ∈ A , The previous maps are well-defined since the images of the generatorsof A are precisely in A (and not in its completion). Indeed, by direct computations makinguse of the explicit form (6.8) of the universal R -matrix and the commutation relations of U q ( sl ), one getsΨ s ( Q ) = Q , Ψ s ( Q ) = Q , Ψ s ( Q ) = Q , Ψ s ( Q ) = Q , Ψ s ( Q ) = Q , Ψ s ( Q ) = Q d . (8.8)and Ψ s ( Q ) = Q , Ψ s ( Q ) = Q , Ψ s ( Q ) = Q , Ψ s ( Q ) = Q , Ψ s ( Q ) = Q u , Ψ s ( Q ) = Q . (8.9)We obtain similarly the actions of Ψ s − i on the generators of A .Since the braided R -matrix satisfies the braided Yang–Baxter equation (6.11), we can showthat the defining relations (8.1) of the braid group B are reproducedΨ s ◦ Ψ s ◦ Ψ s = Ψ s ◦ Ψ s ◦ Ψ s , (8.10a)Ψ s i ◦ Ψ s − i = Ψ s − i ◦ Ψ s i = id. (8.10b)We extend the automorphisms Ψ S to any S ∈ B byΨ S ( X ) = (Ψ g ◦ Ψ g ◦ · · · ◦ Ψ g (cid:96) )( X ) , (8.11)where S is decomposed as S = g g . . . g (cid:96) and g i ∈ { s , s , s − , s − } . Note that the map (8.11)does not depend on the choice of the decomposition of S due to (8.10). Remark 8.1. The realization of the braid group given by Ψ S is not faithful. For example, onecan verify that Ψ ( s s ) = id . This is checked to be true on the intermediate Casimir elementsby making repeated use of (8.8) - (8.9) . It follows that it is also true for any polynomial in thoseelements. Moreover, some elements of A have additional stabilizers, e.g. Ψ s s ( Q ) = ˇ R − ˇ R − Q ˇ R ˇ R = ˇ R − Q ˇ R = Q , (8.12a)Ψ s ( Q ) = Q . (8.12b) Identifying stabilizers of the braid group action on elements of A is easy to do but giving anexhaustive list is harder. Remark 8.2. It was shown in [81] how such a braid group action translates to the q → − limit. This limit of the Askey–Wilson algebra is referred to as the Bannai–Ito algebra. In thatcase, the B braid group action simplifies to an action of the S symmetric group. It is possibleto study more generally the action of the S n symmetric group on the higher rank Bannai-Itoalgebra B ( n ) . Sk iq / (Σ , ) We now present a B group action on the Kauffman bracket skein algebra Sk iq / (Σ , ), denoted ψ S : Sk iq / (Σ , ) → Sk iq / (Σ , ), with S ∈ B . The braid group action rotates the placementof the punctures with respect to each other. 19ere is how it goes. First, the actions ψ s i and ψ s − i on Sk iq / (Σ , ) are defined by theso-called half Dehn twists [71, 82]. The four generators of B act as × × × ψ s = , × × × ψ s − = , × × × ψ s = , × × × ψ s − = , (8.13)where any framed link gets deformed continuously without crossing the punctures as the rota-tions happen. For example, one gets ψ s − ( A ) = ψ s − (cid:16) × × × (cid:17) = (cid:32) × × × (cid:33) = × × × = A , (8.14)and ψ s ( A ) = ψ s (cid:16) × × × (cid:17) = (cid:32) × × × (cid:33) = (cid:16) × × × (cid:17) = A . (8.15) Proposition 8.3. The actions ψ g for g ∈ { s , s , s − , s − } are automorphisms of Sk iq / (Σ , ) .Proof. For any X , Y ∈ Sk iq / (Σ , ) and g ∈ { s , s , s − , s − } , one understands that ψ g ( X · Y ) = ψ g ( X ) · ψ g ( Y ) . (8.16)Indeed, from the way they were defined, the rotations do not add or change crossings. Thus,the Kauffman bracket relations (5.1) that one makes use of to “simplify the crossings” of a givenproduct are unchanged under these rotations. Since the rotations are also defined in order toavoid links crossing punctures, the topological properties (such as which punctures are circledby which links) are preserved. Hence the action ψ g is a homomorphism. Moreover ψ g is anendomorphism because links in Sk iq / (Σ , ) are mapped to other links in Sk iq / (Σ , ), andit is invertible, as rotations can be inverted, thus ψ g is an automorphism.Let S = g g . . . g (cid:96) ∈ B be a decomposition of an element of the braid group on threestrands with g i ∈ { s , s , s − , s − } . We define the automorphism ψ S as follows: ψ S ( X ) = ( ψ g ◦ ψ g ◦ · · · ◦ ψ g (cid:96) )( X ) . (8.17)We use also the definition ψ = id . The previous map (8.17) does not depend on the choiceof the decomposition of S . Indeed, it is straightforward to check that the defining relations ofthe braid group (8.1) are verified on the generators. By the homomorphism property (8.16),it follows that these braid relations are verified for any element of the Kauffman bracket skeinmodule Sk iq / (Σ , ). 20 emark 8.4. More visually complicated loops can always be created by further “twisting” theloops. For example, ψ ( s − ) ( A ) = × × × (8.18) is a more complicated analog of A . The shadow filling the inside of the loop is there to guidethe eyes of the reader. These have also been studied in [83]. Remark 8.5. Let us remark that in [71], the author considers a similar braid group actionby half Dehn twists on the Kauffman bracket skein algebra of the four-punctured sphere. Inthat paper, it is shown that the group SL (2; Z ) acts on the DAHA of type ( C ∨ C ) throughconjugations. Furthermore, the Artin braid group B action on the Kauffman bracket skeinalgebra can be seen as a translation of this SL (2; Z ) action. We also note that Terwilliger hadpresented a B action on both the universal Askey–Wilson algebra and the universal DAHA oftype ( C ∨ , C ) [44]. The following proposition establishes the connections between both braid group actions pre-sented above. Proposition 8.6. The following diagram of isomorphisms A A Sk SkΨ S ψ S φ φ is commutative for any S ∈ B . Here we used the shortened notation Sk ≡ Sk iq / (Σ , ) . Theisomorphisms φ , Ψ S and ψ S are given in (6.16) , (8.11) and (8.17) , respectively.Proof. We can show that this diagram is commutative for all the generators of A and for any S = s i or S = s − i . For example: φ ◦ Ψ s ( Q ) = φ ( Q ) = A = ψ s ( A ) = ψ s ◦ φ ( Q ) . (8.19)A more complicated example is φ ◦ Ψ s ( Q ) = φ ( Q u ) = × × × = ψ s ( A ) = ψ s ◦ φ ( Q ) . (8.20)Since all the maps of the diagram are homomorphisms, the commutativity of the diagram onthe generators of A is enough to prove the proposition for any S ∈ B .21he commutativity of this diagram allows us to identify the conjugation by the braided R -matrix for A as half Dehn twists around the punctures of Sk iq / (Σ , ). In addition, wecan identify easily the elements of the algebra A obtained as an image by Ψ S with a link ofSk iq / (Σ , ). ( n ) algebra Some natural generalizations of the different algebras have previously been introduced andstudied: • the generalized Askey–Wilson algebra aw ( n ) is the algebra generated by { C I | I ⊂{ , , . . . , n }} subject to the relations introduced in Theorems 3.1 and 3.2 of [56]; • the algebra A n is the subalgebra of U q ( sl ) ⊗ n generated by all the intermediate Casimirelements { Q I | I ⊂ { , , . . . , n }} obtained by the repeated action of the coproduct of U q ( sl ); • the centralizer C n is defined by C n = { X ∈ U q ( sl ) ⊗ n (cid:12)(cid:12) [∆ ( n − ( x ) , X ] = 0 , ∀ x ∈ U q ( sl ) } (9.1)where ∆ ( n ) = (∆ ( n − ⊗ id )∆ and ∆ (1) = ∆; • the algebra Sk θ (Σ ,n +1 ) is the Kauffman bracket skein algebra associated to the ( n + 1)-punctured sphere Σ ,n +1 [39]. Let us now associate to each set I ⊆ [1; n ] ≡ { , , . . . , n } a ‘simple’ loop A I of Sk θ (Σ ,n +1 ). We write a set I as I = I ∪ I ∪ · · · ∪ I (cid:96) , where I i aresets of consecutive integers and then we define the ‘simple’ loop A I as: A I = . . . × I ... × I . . . × I (cid:96) . . . (9.2)These simple loops do not bend around, unlike (8.18). They are only extending in thelower half of the plane. In particular, for I = { i, i +1 , . . . , j } , a set of consecutive integers,one gets A I = (cid:32) × . . . × i . . . × j . . . × n (cid:33) = (cid:32) . . . × I . . . (cid:33) (9.3)What is lacking in the previous list is the generalization saw ( n ) of the algebra saw (3).Such a generalization would provide a description of the algebra A n in terms of generatorsand relation. We know that saw ( n ) will be a quotient of the algebra aw ( n ) by relation(s) ofthe type (2.2), with some Casimir elements to be determined. We conjecture that the map φ n from saw ( n ) to Sk θ (Σ ,n +1 ) which sends Q I to A I is an isomorphism † .Let us mention that there also exist generalizations in the non-deformed case ( q = 1 and q = − 1) of the Askey–Wilson algebra: these are respectively called the “higher rank Racahalgebra” introduced in [62] as well as the “higher rank Bannai–Ito algebra” introduced in [63].In the remainder, we give different indications regarding ways to define saw ( n ). † During the preparation of this paper, the authors have been informed by J. Cooke that a similar idea waspursued in an upcoming publication [39]. .1 Punctures on a sphere and a coassociative homomorphism of Kauffmanbracket skein modules Recall we had highlighted that the punctures of the sphere were related to the tensor productfactors. Additionally, a loop encircling a puncture is associated to some intermediate Casimirelement with non-trivial factors in the tensor product factor corresponding to the puncture.Further recall that the coproduct ∆ acts as an algebra morphism from U q ( sl ) to U q ( sl ) ⊗ .One can define an action of the coproduct on any i th factor of a tensor product: for any X ∈ U q ( sl ) ⊗ n , we define ∆ i : U q ( sl ) ⊗ n → U q ( sl ) ⊗ ( n +1) as:∆ i ( x ) = (cid:16) ⊗ ( i − ⊗ ∆ ⊗ ⊗ ( n − i ) (cid:17) ( X ) . (9.4)Now in U q ( sl ) ⊗ some intermediate Casimir elements are related to each other by the coprod-uct, such as Q and Q : ∆ ( Q ) = (∆ ⊗ ⊗ Q = Q ⊗ . (9.5)This relation between Q and Q appears in the framed links picture as well.More precisely, ∆ i has an analog, the δ i morphism, which acts on a single puncture i bycreating another puncture next to it. If the puncture i is enclosed in a loop, the createdpuncture is also enclosed in the same loop. The example (9.5) is illustrated as follows: δ A = δ (cid:16) × × × (cid:17) = δ (cid:16) × × × (cid:17) = (cid:16) × × × × (cid:17) = A ∈ Sk θ (Σ , ) (9.6)This δ i is a Kauffman bracket skein module coassociative algebra homomorphism. It pro-vides embeddings of Sk θ (Σ ,n ) → Sk θ (Σ ,n +1 ). This can be seen as the commutativity of thefollowing diagram: A n A n +1 Sk θ (Σ ,n +1 ) Sk θ (Σ ,n +2 )∆ i δ i φ n φ n +1 The defining algebra relations of Sk iq / (Σ , ) (2.1)–(2.2) (see Proposition 5.2) can be classifiedin three types. The relations always involve two generators, whose product, commutator or q -commutator is reexpressed in terms of other generators. Now imagine we draw both generatorssimultaneously in a framed links diagram (as if we were to multiply them). Some crossingswill appear if the two generators don’t commute. Definition 9.1. The crossing index is defined as the minimal number of crossings that appearin a framed link diagram no matter how the generators are drawn. The relations (2.1)–(2.2) can be classified in terms of the crossing index as follows:23 If the generators can be drawn simultaneously in such a way that the loops have nocrossings (crossing index of 0), they will commute (for example, this is the case for anycentral element Q , Q , Q , Q multiplied with any other generator). • If the generators can be drawn in such a way that their minimum number of crossings istwo (crossing index of 2), linear q -commutation relations of aw (3)-type will be obtained,such as relations (2.1). • If the generators have a crossing index of 4, such as φ ( Q u Q d ) = × × × , (9.7)higher order relations of the type (2.2) will be obtained.This crossing index proves useful for the analysis of the higher rank generalizations of saw (3). ( n ) and Sk θ (Σ ,n +1 ) As mentioned previously, the algebra aw ( n ) is generated by C I with I ⊆ [1; n ] and subject tothe relations of Proposition 3.1 of [56]. We can show by using the action of the morphism δ i thatwe have an homomorphism from aw ( n ) to Sk θ (Σ ,n +1 ). Moreover, we can show that all therelations of Proposition 3.1 of [56] correspond to the product of two simple loops with crossingindex 2. We believe that the relations in [56] exhaust all possibilities of relations involving theproduct of simple loops with crossing index 2. We conjecture also that the above mentionedhomomorphism is surjective (but it is certainly not injective, even for the case n = 3). Thedescription of the kernel would involve products of links with a crossing index strictly greaterthan 2. The complete description of this kernel would lead to the definition of saw ( n ) andgive an algebraic description of A n and Sk θ (Σ ,n +1 ).The study of saw ( n ) should be guided by the intuition gained from the framed links picture.To illustrate the type of insight we can gain, let us efficiently summarize some of the resultsof [53]. In this paper, the authors study the intermediate Casimir elements in U q ( sl ) ⊗ andintroduce an involution I of the algebra as well as “involuted” generators IQ and IQ satisfying [ Q , IQ ] = 0 , and [ IQ , Q ] = 0 . (9.8)That these generators commute becomes evident when we rewrite (following our definitions) IQ = Q u , IQ = Q u , and then draw the corresponding links. Indeed, the products φ ( Q d Q u ) = × × × × = φ ( Q u Q d ) , (9.9a) φ ( Q u Q d ) = × × × × = φ ( Q d Q u ) , (9.9b)24ave 0 crossing hence [ Q d , Q u ] = 0 and [ Q u , Q d ] = 0.What about the product of terms like Q d and Q d ? This calculation has never appearedin the papers mentioned above because it has a crossing number of 4: φ ( Q d Q d ) = × × × × (9.10)Remarkably, this calculation can be effected in Sk iq / (Σ , ) using the conjectured morphism.One writes the Q I in terms of A I , computes using the skein relations of Sk iq / (Σ , ), thenreexpresses all A I in terms of Q I . This yields the following results Q Q = q Q Q + q − Q Q + q ( Q Q Q + Q Q Q ) + q − ( Q Q Q + Q Q Q )+ ( q + q − ) Q + Q Q Q Q + Q Q + Q Q + Q Q + Q Q (9.11)and Q Q = q − Q Q + q Q Q + q − ( Q Q Q + Q Q Q ) + q ( Q Q Q + Q Q Q )+ ( q + q − ) Q + Q Q Q Q + Q Q + Q Q + Q Q + Q Q . (9.12)These have been checked to hold in U q ( sl ) ⊗ .Let us also mention that the action of the braid group B can be generalized to the actionof B n on Sk θ (Σ ,n +1 ) and A n . This might turn out useful for proving results in the future. 10 Conclusion Three objectives were principally pursued in this paper. The first aimed to review the differentavatars of the Askey–Wilson algebra and to clarify the relations between them. Among thosealgebras, we focused on two and presented novel results related to these cases; this was thesecond main goal. The Special Zhedanov algebra sZh q ( m , m , m , m ) was obtained from (aquotient of) the reflection algebra by setting the Sklyanin determinant to an appropriate value;its W ( D ) symmetry was exhibited in addition. The Special Askey–Wilson algebra saw (3),a universal analogue of sZh q ( m , m , m , m ), was shown to be isomorphic to the algebra A that emerges from the Racah problem of U q ( sl ) and also to the Kauffman bracket skein algebraof the four-punctured sphere Sk iq / (Σ , ). An injective homomorphism between A and thecentralizer C of U q ( sl ) in its threefold tensor product was stated and proved. Actions of thebraid group on both Sk iq / (Σ , ) (through half Dehn twists) and A (through conjugation bybraided R -matrices) were illustrated and shown to be compatible. The third main objectivewas to discuss the generalization of saw (3) to saw ( n ). To that end, we emphasized thediagrammatic approach, defined a crossing index, and revisited the results of [53, 56] in aunified fashion.Let us conclude with more remarks regarding generalizations of Askey–Wilson algebras. Itwould certainly be desirable to return to Zhedanov’s original quest and to determine directlyfrom the multivariate Askey–Wilson polynomials (of Tratnik type) [57] the algebra that encap-sulates their bispectral properties. Steps have been carried out [55, 58, 59] but this should becompleted. A definite higher rank generalization of the Zhedanov algebra will emerge, whosequotients and central extensions could then be examined and should connect to various fieldsin mathematics and physics. Considering higher rank Lie algebras g instead of sl is anotheravenue that should be explored. The centralizer of the diagonal action of U q ( g ) in the n -fold25ensor product U q ( g ) ⊗ n , or the algebra generated by all the intermediate Casimir elementsof g in the associated Racah problem should be studied. Connections with a generalizationof Sk θ (Σ ,n ) to punctured manifolds of higher genera would be worth investigating (see also[80]). We may also wonder whether the braided universal R -matrix of U q ( g ) plays a role inthis context. Furthermore, the truncated reflection algebra presented in Section 3 provides anatural framework to obtain generalizations of Zhedanov algebras. Different possibilities arehere conceivable. One could consider more general truncations of the reflection algebra. Thistype of generalization has been already studied in [60] and has been associated to quotientsof q -Onsager algebras ‡ . Connections with centralizers and/or skein algebras remain to beexamined. Another possibility with respect to truncated reflection algebras is the following.Instead of using the R -matrix associated to quantum affine algebras, one could consider the R -matrix corresponding to Yangians. In this case, a particular truncation of the reflectionalgebra leads to the Hahn algebra, which is a specialization of the Zhedanov algebra, see [84].Other truncations should provide interesting generalizations of this algebra. Finally, the FRTpresentation of the reflection algebra associated to higher rank Lie algebras and superalgebrasis well-known. For instance, the twisted Yangians Y tw ( o n ) and Y tw ( sp n ) [85] and the reflectionalgebra B ( n, (cid:96) ) [86] correspond to subalgebras of the Yangian of sl n . Some q -deformations ofthese structures have been also studied previously [87] and are related to the quantum affinealgebra of sl n . Their truncations have yet to be scrutinized and should possess interestingfeatures § . These ideas that we plan on pursuing in the near future are indications that thereis much lying ahead with respect to algebras of the Askey–Wilson type and what they willreveal and lead to. Acknowledgments Many thanks to Geoffroy Bergeron for long-drawn discussions. We have also benefitted fromexchanging with Pascal Baseilhac, Juliet Cooke, Hendrik De Bie, Hadewijch De Clercq, SarahPost, Paul Terwilliger and Alexei Zhedanov. N. Cramp´e and L. Poulain d’Andecy are par-tially supported by Agence Nationale de la Recherche Projet AHA ANR-18-CE40-0001. L.Frappat is grateful to the Centre de Recherches Math´ematiques (CRM) for hospitality andsupport during his visit to Montreal in the course of this investigation. J. Gaboriaud holdsan Alexander-Graham-Bell scholarship from the Natural Sciences and Engineering ResearchCouncil of Canada (NSERC). The research of L. Vinet is funded in part by a Discovery Grantfrom NSERC. A Classical limit and injectivity We provide an explicit description of the classical limit of the realization of saw (3) in U q ( sl ) ⊗ in terms of polarized traces, and use it to prove the injectivity of the map from saw (3) to thecentralizer C . In this appendix, we will work with the formal series version of U q ( sl ) and ‡ The classical limit q → sl and to quotients of the Onsageralgebra by Davis relations [61]. § Such an approach has been pursued in the classical limit q → sl n Onsager algebra [89]. U ( sl ),where we can use known results of classical invariant theory involving polarized traces. A.1 Polarised traces in U ( sl ) ⊗ The algebra U ( sl ) is generated by elements e ij , i, j ∈ { , } , with the defining relations[ e ij , e kl ] = δ jk e il − δ li e kj and e + e = 0. To join up with the notations used in the paperfor U q ( sl ), we set E = e , F = e and H = ( e + e ) = e = − e , and the relationsbecome: [ H, E ] = E, [ H, F ] = − F, [ E, F ] = 2 H. (A.1)In a tensor product U ( sl ) ⊗ N , we denote the generators by e ( a ) ij , where a ∈ { , . . . , N } indicatesthe corresponding factor in the tensor product. The polarized traces are the following elements: T ( a ,...,a d ) = e ( a ) i i e ( a ) i i . . . e ( a d ) i i d , a , . . . , a d ∈ { , . . . , N } , (A.2)where the summation over repeated indices is understood. The specific combinations of polar-ized traces that will appear are: k := T (1 , , k := T (2 , , k := T (3 , , k := k + k + k + 2( T (1 , + T (2 , + T (1 , ) ,X := k + k + 2 T (1 , , Y := k + k + 2 T (2 , , Z := − T (1 , , . (A.3) A.2 The algebra U α ( sl ) In this appendix, we will work with the formal series version of the quantum group U q ( sl ). Weconsider a formal parameter α . The algebra U α ( sl ) is, as a vector space, the space U ( sl )[[ α ]]of all formal power series in α with coefficients in U ( sl ), and the multiplication is determinedby the defining relations of U q ( sl ), see section 6.1, where q is replaced by e α and q H is replacedby e αH . This results in the following relations deforming (A.1):[ H, E ] = E, [ H, F ] = − F, [ E, F ] = e αH − e − αH e α − e − α . (A.4)Similarly, the algebra U α ( sl ) ˆ ⊗ N is the vector space U ( sl ) ⊗ N [[ α ]] of formal series with coef-ficients in U ( sl ) ⊗ N and multiplication induced by the above relations in each factor. Thecomultiplication of U α ( sl ) is naturally obtained from the comultiplication given for U q ( sl ).Note that the limit α → U α ( sl ) yields the algebra U ( sl ) and the comultiplicationbecomes the diagonal embedding. A.3 Reduction to U ( sl ) We want to prove that the following elements Q i Q j Q k Q m Q n Q p Q q , i, j, m, n, p, q ∈ N , k ∈ { , } , (A.5)are linearly independent in U α ( sl ) ˆ ⊗ . First it is more convenient (and equivalent) to replacethe generators Q I by the modified versions introduced Section 4: K I = Q I − ( q + q − )( q − q − ) , I ∈ { , , , , , } . (A.6)27he index 13 does not belong to this set, and for this one, we set: K = Q − ( Q + Q + Q + Q − Q − Q ) + ( q + q − )( q − q − ) . (A.7)Calculating explicitly the first terms in the expansions in α (up to order 3 for Q and up toorder 2 for the others), we find that the new elements K I are well-defined in U α ( sl ) ˆ ⊗ , andmoreover that their degree 0 coefficients are expressed in terms of polarized traces, using thenotations in (A.3), as follows K i | α =0 = 12 k i ( i = 1 , , , K | α =0 = 12 k , (A.8) K | α =0 = 12 X, K | α =0 = 12 Y, K | α =0 = − Z. (A.9)These are straightforward calculations, the one for K being a bit lengthy (for which one canuse for example the explicit expression for Q using the R -matrix given in the paper).Now, to prove that the elements of the set (A.5), with Q I replaced by K I , are linearlyindependent in U α ( sl ) ˆ ⊗ , it is enough to prove that their “classical limits” (the degree 0coefficients) are linearly independent in U ( sl ) ⊗ . In view of the above calculations, it remainsto show that the following set: k i k j k k k m X n Y p Z q , i, j, k, m, n, p ∈ N , q ∈ { , } , (A.10)is linearly independent in U ( sl ) ⊗ . A.4 Racah algebra and diagonal centraliser in U ( sl ) ⊗ To prove that the set (A.10) is linearly independent, we use the same line of arguments as theone used in the study of the recoupling of two copies of sl (3). Thus we only give here a sketchand refer for more details to [77].It is known from classical invariant theory [90, 91] that the centralizer of the diagonalembedding of U ( sl ) in U ( sl ) ⊗ is generated by the polarised traces T ( i,i ) , T ( k,l ) , T (1 , , , with i = 1 , , ≤ k < l ≤ 3, and moreover that the Hilbert–Poincar´e series of the centralizeris: 1 − t (1 − t ) (1 − t ) . (A.11)This series records the dimension for each degree of the centralizer, where the degree in U ( sl ) ⊗ is defined by deg ( e ( a ) ij ) = 1. From this information, we extract at once that the set k , k , k , k , X , Y , Z generates the centralizer. Now, we have that these elements satisfy the classicalRacah relations: k , k ,k , k commute with all generators , [ X, Y ] = Z, [ X, Z ] = 4 { X, Y } + 4 X − k + k + k + k ) X + 4( k − k )( k − k ) , [ Z, Y ] = 4 { X, Y } + 4 Y − k + k + k + k ) Y + 4( k − k )( k − k ) , (A.12)together with Γ = 8( k − k + k − k )( k k − k k ) − k k + k k ) , (A.13)28here the element Γ isΓ : = Z − XY X + Y XY ) + 4( k + k + k + k − { X, Y }− k − k )( k − k ) Y − k − k )( k − k ) X. (A.14)The relations (A.12) allow to rewrite any product in terms of ordered monomials in the gener-ators and (A.13) allows to rewrite Z . So we deduce easily that the set (A.10) is a spanning setfor the centralizer. Finally, the comparison with the Hilbert–Poincar´e series in (A.11) showsthat this set must be linearly independent.This concludes the proof of the injectivity of the map from saw (3) to C ⊂ U α ( sl ) ˆ ⊗ . Remark A.1. 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