Temperley-Lieb, Birman-Murakami-Wenzl and Askey-Wilson algebras and other centralizers of U q ( sl 2 )
aa r X i v : . [ m a t h . QA ] A ug TEMPERLEY–LIEB, BIRMAN–MURAKAMI–WENZL AND ASKEY–WILSONALGEBRAS AND OTHER CENTRALIZERS OF U q ( sl ) NICOLAS CRAMP´E † , LUC VINET ∗ , AND MERI ZAIMI ∗∗ Abstract.
The centralizer of the image of the diagonal embedding of U q ( sl ) in the tensor productof three irreducible representations is examined in a Schur–Weyl duality spirit. The aim is to offera description in terms of generators and relations. A conjecture in this respect is offered withthe centralizers presented as quotients of the Askey–Wilson algebra. Support for the conjecture isprovided by an examination of the representations of the quotients. The conjecture is also shownto be true in a number of cases thereby exhibiting in particular the Temperley–Lieb, Birman–Murakami–Wenzl and one-boundary Temperley–Lieb algebras as quotients of the Askey–Wilsonalgebra. Introduction
The objective of this paper is to establish precisely the connections between the Askey–Wilsonalgebra and the centralizers of the quantum algebra U q ( sl ) such as the Temperley–Lieb and Birman–Murakami–Wenzl algebras.In previous works, the connections between the Racah, Temperley–Lieb and Brauer algebrasand other centralizers of sl were studied in the spirit of the Schur–Weyl duality [4]. In a similarfashion, the Bannai-Ito algebra was connected to the centralizers of the superalgebra osp (1 | q -deformation.The Askey–Wislon algebra was first introduced in [22] and is defined by three generators sat-isfying some q -commutation relations. This algebra encodes the properties of the Askey–Wilsonpolynomials [13] and is related to the Racah problem for U q ( sl ) [8]. Due to this connection, a cen-trally extended Askey–Wilson algebra can be mapped to the centralizer of the diagonal embeddingof U q ( sl ) into U q ( sl ) ⊗ [21, 9]. In the q -deformation of the universal enveloping algebra of sl tothe quantum algebra U q ( sl ), the Askey–Wilson algebra plays a role analogous to that of the Racahalgebra.From the Schur–Weyl duality, the centralizer of the diagonal embedding of U q ( sl ) in the tensorproduct of its fundamental representation is connected to the Hecke algebra. In the case of thethreefold tensor product, the centralizer is known [12] to be isomorphic to the Temperley–Liebalgebra [20], which is a quotient of the Hecke algebra. In fact, the algebra U q ( sl ) has infinitelymany finite irreducible representations, labeled by a half-integer or integer spin j . In the case of thetensor product of three spin-1 representations, it is also known [16] that the centralizer of U q ( sl )is isomorphic to the Birman–Murakami–Wenzl algebra [11], which is a q -deformation of the Braueralgebra. However, in the general case of three irreducible representations of spins j , j and j , analgebraic description of the centralizer is not known. The present paper provides an attempt to describe the centralizer of the image of the diagonalembedding of U q ( sl ) in the tensor product of any three irreducible representations in terms ofgenerators and relations, by using the connections with the Askey–Wilson algebra. It is first shownthat there is a surjective map (between generators) from the Askey–Wilson algebra to the centralizer.This statement corresponds in invariant theory to the first fundamental theorem [14]. A conjectureis then proposed in order to obtain an isomorphism between a quotient (given in terms of relations)of the Askey–Wilson algebra and the centralizer of U q ( sl ) – this relates to the second fundamentaltheorem in invariant theory [14]. The conjecture is proved for three spin- representations, inwhich case the Temperley–Lieb algebra is obtained explicitly as a quotient of the Askey–Wilsonalgebra. Similarly, for three spin-1 representations, it is shown that the conjecture holds and thatthe Birman–Murakami–Wenzl algebra is isomorphic to a quotient of the Askey–Wilson algebra.The conjecture is also verified for three spin- representations, and for one spin- j and two spin- representations, for j any spin greater than . In the latter case, it is shown that the centralizer isisomorphic to the one-boundary Tempereley–Lieb algebra [17, 18, 19].The plan of this paper is as follows. Section 2 gives the precise connection between the centralizerof U q ( sl ) and the Askey–Wilson algebra. Subsection 2.1 presents the quantum algebra U q ( sl ) andits properties. The centralizer of U q ( sl ) in U q ( sl ) ⊗ and the intermediate Casimirs are defined inSubsection 2.2. A homomorphism between the centrally extended Askey–Wilson algebra AW (3)and this centralizer is given in Subsection 2.3. Section 3 is concerned with the representations ofelements in U q ( sl ) ⊗ . The finite irreducible representations of U q ( sl ) and their tensor productdecomposition rules are recalled in Subsection 3.1. Subsection 3.2 introduces the object of maininterest, that is the centralizer of the image of the diagonal embedding of U q ( sl ) in the tensorproduct of three irreducible representations. Section 4 aims to describe this centralizer in terms ofgenerators and relations. Subsection 4.1 maps AW (3) to this centralizer, and this is shown to be asurjection in Subsection 4.2. The kernel of this map is discussed in Subsection 4.3, and a conjectureproposing that a quotient of AW (3) is isomorphic to the centralizer is formulated. Subsection 4.4contains the proof that the conjecture does not depend on the ordering of the three spins j , j , j .In order to support the conjecture, Section 5 studies the finite irreducible representations of thequotient of AW (3). The remaining sections contain the proofs of the conjecture for some particularcases. Section 6 focuses on the case j = j = j = . It is shown in Subsection 6.1 that theconjecture holds in this case, and the precise connection with the Temperley–Lieb algebra is givenin Subsection 6.2. Section 7 considers the case j = j = j = 1. The proof of the conjecture is givenin Subsection 7.1. An isomorphism between the quotient of AW (3) and the Birman–Murakami–Wenzl algebra is obtained in Subsection 7.2. The conjecture for the case j = j = j = is provedin Section 8. Finally, Section 9 studies the case j = j for j = 1 , , ... and j = j = . Theconjecture is verified in Subsection 9.1, and the connection with the one-boundary Temperley–Liebalgebra is described in Subsection 9.2.2. Centralizer of U q ( sl ) and Askey–Wilson algebra In this section, we recall well-known properties of the quantum algebra U q ( sl ) to fix the notations.Then, the definition of the centralizer of the diagonal embedding of U q ( sl ) in U q ( sl ) ⊗ is recalledand its homomorphism with the centrally extended Askey–Wilson algebra AW (3) is presented. SKEY–WILSON ALGEBRA AND CENTRALIZERS 3 U q ( sl ) algebra. The associative algebra U q ( sl ) is generated by E , F and q H with the definingrelations(2.1) q H E = qEq H , q H F = q − F q H and [ E, F ] = [2 H ] q , where [ X ] q = q X − q − X q − q − . Throughout this paper, q is a complex number not root of unity. There is acentral element in U q ( sl ), called quadratic Casimir element, given by(2.2) Γ = ( q − q − ) F E + qq H + q − q − H . There exists also an algebra homomorphism ∆ : U q ( sl ) → U q ( sl ) ⊗ U q ( sl ), called comultiplica-tion, defined on the generators by(2.3) ∆( E ) = E ⊗ q − H + q H ⊗ E , ∆( F ) = F ⊗ q − H + q H ⊗ F and ∆( q H ) = q H ⊗ q H . This comultiplication is coassociative(2.4) (∆ ⊗ id)∆ = (id ⊗ ∆)∆ =: ∆ (2) . We define the opposite comultiplication ∆ op = σ ◦ ∆, where σ ( x ⊗ y ) = y ⊗ x , for x, y ∈ U q ( sl ).It is a homomorphism from U q ( sl ) to U q ( sl ) ⊗ U q ( sl ) different from ∆. Both are related by theuniversal R -matrix R ∈ U q ( sl ) ⊗ U q ( sl ) satisfying(2.5) ∆( x ) R = R ∆ op ( x ) for x ∈ U q ( sl ) . The universal R -matrix also satisfies the Yang-Baxter equation(2.6) R R R = R R R . We have used the usual notations: if R = R α ⊗ R α , then R = R α ⊗ R α ⊗ R = 1 ⊗ R α ⊗ R α and R = R α ⊗ ⊗ R α (the sum w.r.t. α is understood).2.2. Centralizer of U q ( sl ) in U q ( sl ) ⊗ . The centralizer C of the diagonal embedding of U q ( sl )in U q ( sl ) ⊗ is(2.7) C = { X ∈ U q ( sl ) ⊗ (cid:12)(cid:12) [∆ (2) ( x ) , X ] = 0 , ∀ x ∈ U q ( sl ) } . This centralizer is a subalgebra of U q ( sl ) ⊗ and we want to describe this subalgebra with somegenerators and defining relations. Let us first give some elements of C by using the Casimirelement Γ which is central in U q ( sl ). We define the following Casimir elements of U q ( sl ) ⊗ (2.8) Γ = Γ ⊗ ⊗ , Γ = 1 ⊗ Γ ⊗ , Γ = 1 ⊗ ⊗ Γ . These elements are central in U q ( sl ) ⊗ and thus belong to C . We also define the total Casimir(2.9) Γ = ∆ (2) (Γ) . This element belongs to C because [∆ (2) (Γ) , ∆ (2) ( x )] = ∆ (2) ([Γ , x ]) = 0 for all x ∈ U q ( sl ). Let usnotice that Γ is central in C since it is also an element of the diagonal embedding of U q ( sl ).We then define the intermediate Casimirs associated to the recoupling of the two first or the twolast factors of U q ( sl ) ⊗ (2.10) Γ = ∆(Γ) ⊗ = 1 ⊗ ∆(Γ) . N.CRAMP´E, L.VINET, AND M.ZAIMI
One uses the properties of the comultiplication to show that Γ and Γ are in C ; indeed, for all x ∈ U q ( sl ), [Γ , ∆ (2) ( x )] = [∆(Γ) ⊗ , (∆ ⊗ id)∆( x )] = (∆ ⊗ id)[Γ ⊗ , ∆( x )] = 0 , (2.11) [Γ , ∆ (2) ( x )] = [1 ⊗ ∆(Γ) , (id ⊗ ∆)∆( x )] = (id ⊗ ∆)[1 ⊗ Γ , ∆( x )] = 0 . (2.12)In the limit q →
1, it can be shown that the element(2.13) Γ = X α Γ α ⊗ ⊗ Γ α , where ∆(Γ) = P α Γ α ⊗ Γ α , belongs to the centralizer of U ( sl ) in U ( sl ) ⊗ . However, this is not thecase for the quantum algebra U q ( sl ). This difficulty that arises in the q -deformation of the algebra U ( sl ) was addressed in [2] where a definition of the third intermediate Casimir element of U q ( sl )is provided with the help of the universal R -matrix. It is shown in [2] that the following elementsΓ (0)13 = R Γ R − = R − Γ R , (2.14) Γ (1)13 = R Γ R − = R − Γ R , (2.15)are in the centralizer C .2.3. Connection with the Askey–Wilson algebra.
The intermediate Casimir elements Γ ,Γ , Γ (0)13 and Γ (1)13 do not commute pairwise but satisfy certain relations which are identified as thoseof the Askey–Wilson algebra AW (3). Definition 2.1.
The centrally extended Askey–Wilson algebra AW (3) is generated by A , B , D andcentral elements α , α , α and K subject to the following defining relations A + [ B, D ] q q − q − = α α + α Kq + q − , (2.16) B + [ D, A ] q q − q − = α α + α Kq + q − , (2.17) D + [ A, B ] q q − q − = α α + α Kq + q − , (2.18) where [ X, Y ] q = qXY − q − Y X . We also define the element D ′ ∈ AW (3) by the following relation D ′ + [ B, A ] q q − q − = α α + α Kq + q − . (2.19)The algebra AW (3) has a Casimir element given by(2.20) Ω = qA ( α α + α K )+ q − B ( α α + α K )+ qD ( α α + α K ) − q A − q − B − q D − qABD . The connection between the centralizer C defined by (2.7) and the Askey–Wilson algebra is givenin the following proposition. Proposition 2.1.
The map ϕ : AW (3) → C defined by ϕ ( α i ) = Γ i , ϕ ( A ) = Γ , ϕ ( B ) = Γ , ϕ ( K ) = Γ , (2.21) is an algebra homomorphism. We deduce that ϕ ( D ) = Γ (0)13 , ϕ ( D ′ ) = Γ (1)13 . (2.22) SKEY–WILSON ALGEBRA AND CENTRALIZERS 5
The homomorphism has been proved in [8]; a direct computation shows that the intermediateCasimir elements satisfy all the relations of AW (3). Relations (2.22) have been proved more recentlyin [2] and a simpler proof of the homomorphism using the universal R -matrix has also been given.Let us remark that a similar proof has also been simplified in the case of the Bannai–Ito algebraand the centralizer for the super Lie algebra osp (1 |
2) [6].Using (2.18) to replace D in (2.16) and (2.17), one shows that the following relations provide anequivalent presentation of AW (3) which will be useful for later computations[ B, [ A, B ] q ] q ( q − q − ) =( q + q − ) A + ( α α + α K ) B − ( q + q − )( α α + α K ) , (2.23) [[ A, B ] q , A ] q ( q − q − ) =( q + q − ) B + ( α α + α K ) A − ( q + q − )( α α + α K ) . (2.24)Furthermore, noticing that [ X, [ Y, X ] q ] q = [[ X, Y ] q , X ] q and using the element D ′ defined in (2.19),one finds that (2.23) and (2.24) imply A + [ D ′ , B ] q q − q − = α α + α Kq + q − , (2.25) B + [ A, D ′ ] q q − q − = α α + α Kq + q − . (2.26)Relations (2.19), (2.25) and (2.26) provide another Z symmetric presentation of AW (3). Remark 2.1.
Upon performing the affine transformation X = ( q − q − ) ˜ X + q + q − on the elements X = A, B, D, D ′ , α i , K of AW (3) , one sees that relations (2.23) – (2.24) can be written as [ ˜ B, [ ˜ A, ˜ B ] q ] q =( q + q − ) (cid:16) − ˜ B − { ˜ A, ˜ B } + ( ˜ K + ˜ α + ˜ α + ˜ α ) ˜ B + ( ˜ α − ˜ K )( ˜ α − ˜ α ) (cid:17) (2.27) +( q − q − ) ( ˜ α ˜ α + ˜ α ˜ K ) ˜ B , [[ ˜ A, ˜ B ] q , ˜ A ] q =( q + q − ) (cid:16) − ˜ A − { ˜ A, ˜ B } + ( ˜ K + ˜ α + ˜ α + ˜ α ) ˜ A + ( ˜ α − ˜ K )( ˜ α − ˜ α ) (cid:17) (2.28) +( q − q − ) ( ˜ α ˜ α + ˜ α ˜ K ) ˜ A , where { X, Y } = XY + Y X . By taking the limit q → of (2.27) and (2.28) , one recovers thedefining relations of the Racah algebra used in [4] . Relations (2.18) and (2.19) are transformed into [ ˜ A, ˜ B ] q q − q − = ˜ α ˜ α + ˜ α ˜ K + q + q − ( q − q − ) ( ˜ α + ˜ α + ˜ α + ˜ K − ˜ A − ˜ B − ˜ D ) , (2.29) [ ˜ B, ˜ A ] q q − q − = ˜ α ˜ α + ˜ α ˜ K + q + q − ( q − q − ) ( ˜ α + ˜ α + ˜ α + ˜ K − ˜ A − ˜ B − ˜ D ′ ) . (2.30) In the limit q → , the elements ˜ D and ˜ D ′ are equal, and the images by ϕ of (2.29) and (2.30) both reduce to the well-known linear relation ˜Γ + ˜Γ + ˜Γ + ˜Γ − ˜Γ − ˜Γ − ˜Γ = 0 that holdsin U ( sl ) ⊗ . Decomposition of tensor product of representations and centralizer
In the previous section, we introduced the centralizer C of the diagonal embedding of U q ( sl ) in U q ( sl ) ⊗ and showed the connection of this subalgebra of U q ( sl ) ⊗ with the Askey–Wilson algebra N.CRAMP´E, L.VINET, AND M.ZAIMI AW (3). We now focus on the corresponding objects when each U q ( sl ) in U q ( sl ) ⊗ is taken in afinite irreducible representation.3.1. Finite irreducible representations of U q ( sl ) . The quantum algebra U q ( sl ) has finite ir-reducible representations of dimension 2 j + 1 that we will denote by M j , with j = 0 , , , , ... The representation map will be denoted by π j : U q ( sl ) → End( M j ). We will use the name spin- j representation to refer to M j . The representation of the Casimir element (2.2) in the space M j is(3.1) π j (Γ) = χ j I j +1 where χ j = q j +1 + q − j − , and I j +1 is the 2 j + 1 by 2 j + 1 identity matrix. We define the following sets, for three half-integersor integers j , j and j J ( j , j ) = {| j − j | , | j − j | + 1 , ..., j + j } , (3.2) J ( j , j , j ) = [ j ∈J ( j ,j ) J ( j, j ) . (3.3)Notice that there are no repeated numbers in J ( j , j , j ), and this set is invariant under anypermutation of j , j and j .For q not a root of unity, the tensor product of two irreducible representations of U q ( sl ) decom-poses into the following direct sum of irreducible representations(3.4) M j ⊗ M j = M j ∈J ( j ,j ) M j . Similarly, the threefold tensor product of irreducible representations of U q ( sl ) decomposes into thefollowing direct sum(3.5) M j ⊗ M j ⊗ M j = M j ∈J ( j ,j ,j ) d j M j , where d j ∈ Z > is the degeneracy of M j and is referred to as the Littlewood–Richardson coefficient.3.2. Centralizer of U q ( sl ) in End ( M j ⊗ M j ⊗ M j ) . From now on, we fix three half-integersor integers j , j and j . The centralizer C j ,j ,j of the diagonal embedding of U q ( sl ) in End( M j ⊗ M j ⊗ M j ) is(3.6) C j ,j ,j = { m ∈ End( M j ⊗ M j ⊗ M j ) (cid:12)(cid:12) [ π j ,j ,j (∆ (2) ( x )) , m ] = 0 , ∀ x ∈ U q ( sl ) } , where we have used the shortened notation π j ,j ,j = π j ⊗ π j ⊗ π j . This centralizer as a subalgebraof End( M j ⊗ M j ⊗ M j ) is the object of interest of this paper. In the next section, we conjecturea presentation of this centralizer in terms of generators and relations by using the connections withthe Askey–Wilson algebra AW (3). Let us first recall some known properties of this centralizer.The knowledge of the centralizer permits to write the decomposition rule (3.5) as follows(3.7) M j ⊗ M j ⊗ M j = M j ∈J ( j ,j ,j ) M j ⊗ V j , SKEY–WILSON ALGEBRA AND CENTRALIZERS 7 where V j is a finite irreducible representation of dimension d j of C j ,j ,j . The set { V j | j ∈J ( j , j , j ) } is the complete set of non-equivalent irreducible representations of C j ,j ,j . In par-ticular, one deduces that the dimension of the centralizer is(3.8) dim( C j ,j ,j ) = X j ∈J ( j ,j ,j ) d j . These representations V j are explicitly given in Subsection 4.2.We now define the images of the centralizing elements (2.8)–(2.10) and (2.14)–(2.15) of U q ( sl ) ⊗ in the representation End( M j ⊗ M j ⊗ M j ) as follows(3.9) π j ,j ,j : C → C j ,j ,j Γ i , Γ ij , Γ C i , C ij , C . Therefore, C j ,j ,j contains the elements C i , C ij and C . According to (3.1), the elements C i aresimply constant matrices of value χ j i for i = 1 , ,
3. The intermediate Casimirs C , C , C (0)13 and C (1)13 and the total Casimir C of C j ,j ,j can be diagonalized if q is not a root of unity.Since C is the Casimir associated to the recoupling of the two first factors of the threefoldtensor product of U q ( sl ), one finds (using the decomposition rule (3.4)) that its eigenvalues are χ j for j ∈ J ( j , j ). Similarly, the eigenvalues of C (resp. C ) are χ j for j ∈ J ( j , j ) (resp. J ( j , j , j )). The same argument cannot be applied directly to the intermediate Casimirs C (0)13 and C (1)13 since they are not trivial in the space 2. However, the element C defined in (2.13) onlycouples the spaces 1 and 3 such that its eigenvalues are χ j for j ∈ J ( j , j ). From the definitions(2.14) and (2.15), we see that C (0)13 and C (1)13 are both conjugations of C by an R -matrix. Hence,their eigenvalues are the same as those of C . The previous discussion implies that the minimalpolynomials of the intermediate Casimirs and the total Casimir take the following form Y j ∈J ( j ,j ) ( C − χ j ) = 0 , Y j ∈J ( j ,j ( C − χ j ) = 0 , Y j ∈J ( j ,j ,j ) ( C − χ j ) = 0 , (3.10) Y j ∈J ( j ,j ) ( C (0)13 − χ j ) = 0 , Y j ∈J ( j ,j ) ( C (1)13 − χ j ) = 0 . (3.11)Because C is central in C j ,j ,j , it can be diagonalized simultaneously with C , C , C (0)13 or C (1)13 . Therefore, one gets the following minimal polynomials Y m ∈M ( j ,j ,j ) ( C − C − m ) = 0 , Y m ∈M ( j ,j ,j ) ( C − C − m ) = 0 , (3.12) Y m ∈M ( j ,j ,j ) ( C − C (0)13 − m ) = 0 , Y m ∈M ( j ,j ,j ) ( C − C (1)13 − m ) = 0 , (3.13)where(3.14) M ( j a , j b , j c ) = [ j ∈J ( j a ,j b ) { χ ℓ − χ j (cid:12)(cid:12) ℓ ∈ J ( j, j c ) } . In the previous set M ( j a , j b , j c ), there are no repeated numbers.Before concluding this section, let us notice that if one performs the transformation given inRemark 2.1 on the Casimir element Γ of U q ( sl ), its value in the representation End( M j ) is ˜ χ j =[ j ] q [ j + 1] q . By construction, similar results hold for the eigenvalues of the transformed elements N.CRAMP´E, L.VINET, AND M.ZAIMI ˜ C i , ˜ C ij and ˜ C . In the limit q →
1, the minimal polynomials of these transformed elements thusreduce to the ones discussed in [4].4.
Algebraic description of the centralizer C j ,j ,j Take j , j and j to be three fixed half-integers or integers. This section contains an attemptto give a definition of the centralizer C j ,j ,j in terms of generators and relations. We rely on theconnection with the Askey–Wilson algebra AW (3).4.1. Homomorphism with AW (3) . The intermediate Casimir elements C i , C ij and C belong-ing to C j ,j ,j satisfy the defining relations of the Askey–Wilson algebra as stated precisely in thefollowing proposition. Proposition 4.1.
The map φ : AW (3) → C j ,j ,j defined by φ ( α i ) = C i , φ ( A ) = C , φ ( B ) = C , φ ( K ) = C , (4.1) is an algebra homomorphism.Proof. The result follows from the fact that φ is the composition of two homomorphisms φ = π j ,j ,j ◦ ϕ , where ϕ is defined in Proposition 2.1. (cid:3) We recall that C i is χ j i = q j i +1 + q − j i − times the identity matrix. Therefore, it can be identifiedas the number χ j i . Let us also emphasize that φ ( D ) = C (0)13 and φ ( D ′ ) = C (1)13 . Moreover, the imageby φ of the Casimir element Ω of AW (3) defined in (2.20) is equal to an expression involving onlycentral elements [8] :(4.2) φ (Ω) = C + C + C + C + C C C C − ( q + q − ) . Surjectivity.
We now show that the intermediate Casimir elements C i , C , C and C generate the whole centralizer C j ,j ,j . Proposition 4.2.
The map φ : AW (3) → C j ,j ,j is surjective.Proof. To reach that conclusion, we prove that the dimension of the image of φ is at least X ℓ ∈J ( j ,j ,j ) d ℓ ,the dimension of C j ,j ,j . Let ℓ ∈ J ( j , j , j ) and(4.3) S ℓ ( j , j , j ) = { j ∈ J ( j , j ) | ℓ ∈ J ( j, j ) } . From the definition (3.2), we deduce that S ℓ ( j , j , j ) = { j min , j min + 1 , . . . , j max } with(4.4) j min = max( | j − j | , | j − ℓ | ) and j max = min( j + j , j + ℓ ) . The cardinality of this set is d ℓ = j max − j min + 1. We denote by M + ℓ the vector space spanned bythe highest weight vectors of the representations M ℓ in the decomposition (3.5). The dimension of M + ℓ is d ℓ and we can choose d ℓ independent vectors v j ∈ M + ℓ with j ∈ S ℓ ( j , j , j ) such that(4.5) π j ,j ,j (∆ (2) ( E )) v j = 0 , π j ,j ,j (∆ (2) ( q H )) v j = q ℓ v j , C v j = χ ℓ v j , C v j = χ j v j , and(4.6) C v j = X k ∈S ℓ ( j ,j ,j ) α j,k v k , SKEY–WILSON ALGEBRA AND CENTRALIZERS 9 where α j,k are complex numbers. The elements C and C are the images by φ of A and B .Therefore, they satisfy the Askey–Wilson algebra. It is enough to determine the constants α j,k asshown previously in [22]. We reproduce this computation in the particular case needed here. Wedefine the constants(4.7) a = χ j χ j + χ j χ ℓ , b = χ j χ j + χ j χ ℓ and c = χ j χ j + χ j χ ℓ . We act with relation (2.24) on the vector v j (for j ∈ S ℓ ( j , j , j )) and project the result on v k with k = j and on v j . We get (cid:16) [ j + k + 2] q [ j + k ] q [ k − j − q [ k − j + 1] q (cid:17) α j,k = 0 , (4.8) α j,j = c χ j − b χ χ j − χ for j = 0 . (4.9)The projection on v j is trivial if j = 0. From relation (4.8), we deduce that α j,k = 0 for j ∈S ℓ ( j , j , j ) and k = j + 1 , j − , j .Then, we act with relation (2.23) on the vector v j and project the result on v j − , v j − , . . . , v j +2 .The projections are trivial except the one on v j which gives the following relation(4.10) [2 j + 3] q α j,j +1 α j +1 ,j − [2 j − q α j − ,j α j,j − = 1 χ ( c − χ j α j,j ) α j,j + χ χ j − a , with the boundary conditions α j min ,j min − = 0 and α j max ,j max +1 = 0. By using (4.9), one can showthat the recurrence relation (4.10) and the boundary condition α j max ,j max +1 = 0 imply(4.11) α j − ,j α j,j − = Q i =1 ([ j − r i ] q [ j + r i ] q )[2 j − q [2 j ] q [2 j + 1] q ( q − q − ) for j = 0 , where r = j − j , r = j − ℓ , r = j + j + 1 and r = ℓ + j + 1. We see from (4.4) that thesecond boundary condition α j min ,j min − = 0 is automatically satisfied if j min >
0. In the case where j min = 0 (which only happens if j = j and ℓ = j ), the limit j → α , = χ j χ j /χ , which is the limit j → j min < j ≤ j max , and that the eigenvalues of C are pairwise distinct. Therefore, fora given ℓ ∈ J ( j , j , j ), C and C generate a vector space of dimension d ℓ . (cid:3) Kernel.
The map φ defined in the Proposition 4.1 is not injective since there are non-trivialelements of AW (3) that are mapped to zero, as seen from the results (3.10)–(3.13). We want toprovide a description of the kernel of the map φ in order to find a quotient of AW (3) that isisomorphic to the centralizer C j ,j ,j . Let us first define a quotient of AW (3). Definition 4.1.
The algebra AW ( j , j , j ) is the quotient of the centrally extended Askey–Wilsonalgebra AW (3) by the following relations α i = χ j i , (4.12) Y j ∈J ( j ,j ) ( A − χ j ) = 0 , Y j ∈J ( j ,j ) ( B − χ j ) = 0 , Y j ∈J ( j ,j ,j ) ( K − χ j ) = 0 , (4.13) Y j ∈J ( j ,j ) ( D − χ j ) = 0 , Y j ∈J ( j ,j ) ( D ′ − χ j ) = 0 , (4.14) Y m ∈M ( j ,j ,j ) ( K − A − m ) = 0 , Y m ∈M ( j ,j ,j ) ( K − B − m ) = 0 , (4.15) Y m ∈M ( j ,j ,j ) ( K − D − m ) = 0 , Y m ∈M ( j ,j ,j ) ( K − D ′ − m ) = 0 , (4.16) Ω = χ j + χ j + χ j + K + χ j χ j χ j K − ( q + q − ) , (4.17) where we recall that D and D ′ are defined through (2.18) – (2.19) , and Ω is defined in (2.20) . Let us emphasize that all the relations (4.12)–(4.17) are in the kernel of the map φ in view of theresults of Subsections 3.2 and 4.1. We are now in position to state a conjecture that proposes analgebraic description of C j ,j ,j . Conjecture 4.1.
The map φ : AW ( j , j , j ) → C j ,j ,j given by φ ( A ) = C , φ ( B ) = C , φ ( K ) = C , (4.18) is an algebra isomorphism. To support this conjecture, we remark that by taking the limit q → AW (3)is new and will be useful when we discuss the representations of AW ( j , j , j ) in Section 5.From the previous results, we know that φ is a surjective homomorphism. It remains to provethat it is injective, which can be done by demonstrating that(4.19) dim( AW ( j , j , j )) ≤ X j ∈J ( j ,j ,j ) d j = dim( C j ,j ,j ) . To simplify the demonstration of (4.19), we can decompose AW ( j , j , j ) into a direct sum ofsimpler algebras. Indeed, let us introduce the following central idempotents, for k ∈ J ( j , j , j ),(4.20) K k = Y r ∈J ( j ,j ,j r = k K − χ r χ k − χ r , which satisfy K k K ℓ = δ k,ℓ K k , X k ∈J ( j ,j ,j ) K k = 1 and K K k = K k K = χ k K k . We deduce that(4.21) AW ( j , j , j ) = M k ∈J ( j ,j ,j ) K k AW ( j , j , j ) K k . SKEY–WILSON ALGEBRA AND CENTRALIZERS 11
Then, confirming inequality (4.19) amounts to proving the following inequalities, for k ∈ J ( j , j , j ),(4.22) dim( K k AW ( j , j , j ) K k ) ≤ d k . The algebras K k AW ( j , j , j ) K k are simpler to study than AW ( j , j , j ). Roughly speaking, theycorrespond to replacing in the defining relations of AW ( j , j , j ) the central elements α i (resp. K )by χ j i (resp. χ k ). One thus gets two annihilating polynomials for A (similarly for B , D and D ′ )(4.23) Y j ∈J ( j ,j ) ( A − χ j ) = 0 , Y m ∈M ( j ,j ,j ) ( A − χ k + m ) = 0 , which reduce to only one, i.e. (4.24) Y j ∈J k ( j ,j ,j ) ( A − χ j ) = 0 , with J k ( j a , j b , j c ) = { j ∈ J ( j a , j b ) | χ j ∈ { χ k − m | m ∈ M ( j a , j b , j c ) }} .In fact, in the quotient of C j ,j ,j where C = χ k , the minimal polynomial of C is(4.25) Y j ∈S k ( j ,j ,j ) ( C − χ j ) = 0 , where we recall that (see proof of Proposition 4.2)(4.26) S k ( j a , j b , j c ) = { j ∈ J ( j a , j b ) | k ∈ J ( j, j c ) } . Similar results hold for C , C (0)13 and C (1)13 . Let us emphasize that(4.27) S k ( j a , j b , j c ) ⊆ J k ( j a , j b , j c ) , and that the cardinality of S k ( j a , j b , j c ) is equal to d k and does not depend on the ordering of j a , j b , j c . This discussion suggests the definition of another quotient of AW (3). Definition 4.2.
The algebra AW k ( j , j , j ) , where k ∈ J ( j , j , j ) , is the quotient of the centrallyextended Askey–Wilson algebra AW (3) by α i = χ j i and the following relations K = χ k , (4.28) Y j ∈S k ( j ,j ,j ) ( A − χ j ) = 0 , Y j ∈S k ( j ,j ,j ) ( B − χ j ) = 0 , (4.29) Y j ∈S k ( j ,j ,j ) ( D − χ j ) = 0 , Y j ∈S k ( j ,j ,j ) ( D ′ − χ j ) = 0 . (4.30)Let us remark that the four annihilating polynomials (4.29)–(4.30) are of degree d k . These quotientslead to another conjecture. Conjecture 4.2.
The direct sum (4.31) g AW ( j , j , j ) = M k ∈J ( j ,j ,j ) AW k ( j , j , j ) is isomorphic to C j ,j ,j . As for Conjecture 4.1, the proof of this conjecture reduces to showing that(4.32) dim (cid:16) AW k ( j , j , j ) (cid:17) ≤ d k . In view of (4.27), we see that Conjecture 4.2 is true if Conjecture 4.1 is. Moreover, in this case, AW k ( j , j , j ) is isomorphic to K k AW ( j , j , j ) K k .To conclude this section, let us emphasize that both conjectures presented above would providean algebraic description of the centralizer C j ,j ,j . A strategy to prove these conjectures wouldbe to establish inequalities (4.32) and then to derive the isomorphism between AW k ( j , j , j ) and K k AW ( j , j , j ) K k .4.4. Invariance under permutations of { j , j , j } . The algebras involved in Conjecture 4.1depend on the choice of three spins j , j and j . We now show that it is sufficient to check theconjecture for only one ordering for the spins j , j and j . Proposition 4.3.
Let j , j and j be three positive half-integers or integers. If Conjecture 4.1 istrue for the sequence of spins { j , j , j } , then it is also true for every permutation of j , j , j .Proof. For any two representation maps π j and π j of U q ( sl ), it is known that there exists aninvertible matrix P such that for all x ∈ U q ( sl ) we have ( π j ⊗ π j )(∆( x )) = P − ( π j ⊗ π j )(∆( x )) P .Therefore, from the definition of the centralizer (3.6) and the coassociativity of the comultiplication(2.4), we deduce that for any permutation σ of the symmetric group S , C j ,j ,j is isomorphic to C j σ (1) ,j σ (2) ,j σ (3) .We must now show that the quotiented Askey–Wilson algebra AW ( j σ (1) , j σ (2) , j σ (3) ) is isomorphicto AW ( j , j , j ) for any permutation σ ∈ S . Since S is generated by the transpositions (1 , , φ : AW ( j , j , j ) → AW ( j , j , j ) φ ( α ) = α , φ ( α ) = α , φ ( α ) = α , φ ( A ) = B, φ ( B ) = A, φ ( K ) = K , (4.33) φ : AW ( j , j , j ) → AW ( j , j , j ) φ ( α ) = α , φ ( α ) = α , φ ( α ) = α , φ ( A ) = A, φ ( B ) = D ′ , φ ( K ) = K . (4.34)To see the homomorphism for the defining relations of AW (3), it is easier to work with the symmetricpresentations (2.16)–(2.19) and (2.25)–(2.26). By noticing that φ ( D ) = D ′ and φ ( D ) = B , thehomomorphism immediately follows. In order to preserve relation (4.12), the central elements α i have to be permuted in the same way as the spins j i , which is the case for the two maps givenabove. For the quotiented relations (4.13)–(4.16), the homomorphism is checked by observingthat J ( j a , j b ) = J ( j b , j a ), M ( j a , j b , j c ) = M ( j b , j a , j c ) and that J ( j a , j b , j c ) is invariant under anypermutation of its entries. The R.H.S. of relation (4.17) is invariant under any permutation of theelements α i . By using relations (2.16)–(2.19) and (2.25)–(2.26), it is straightforward to show that φ (Ω) = Ω and φ (Ω) = Ω, which proves the homomorphism for relation (4.17). Finally, since themaps φ and φ are surjective and invertible, they are bijective. (cid:3) SKEY–WILSON ALGEBRA AND CENTRALIZERS 13 Finite irreducible representations of AW ( j , j , j )To support Conjecture 4.1, we want to show that the sum of the squares of the dimensions of allthe finite irreducible representations of AW ( j , j , j ) is equal to the dimension of the centralizer.This result implies that Conjecture 4.1 is true if and only if AW ( j , j , j ) is semisimple. Moreover,if AW ( j , j , j ) is not semisimple, the previous result proves that the missing relations (if there areany) in the kernel of φ are in a nilpotent radical of AW ( j , j , j ).To identify all the finite irreducible representations of AW ( j , j , j ), we use the classificationof the representations of the universal Askey–Wilson algebra given in [10] and look for the oneswhere the different relations of the quotient are satisfied. The universal Askey–Wilson algebra ∆ q ,introduced in [21], is generated by three elements A, B, C and has three central elements α, β, γ .There is a surjective algebra homomorphism from ∆ q to the quotient of AW (3) by α i = χ j i , withthe following mappings A A , B B , C D , (5.1) α χ j χ j + χ j K , β χ j χ j + χ j K, γ χ j χ j + χ j K . (5.2)We deduce that the quotient of AW (3) by α i = χ j i is isomorphic to the quotient of ∆ q by the rela-tions ( α − χ j χ j ) /χ j = ( β − χ j χ j ) /χ j = ( γ − χ j χ j ) /χ j . We can therefore use the representationtheory of ∆ q in order to determine all the finite irreducible representations of AW ( j , j , j ).The finite irreducible modules of the universal Askey–Wilson algebra ∆ q for q not a root of unityare classified in [10]. They are given by the isomorphism classes of the n + 1-dimensional modules V n ( a, b, c ) defined in [10], for n ≥
0, under certain conditions on a, b, c (see Theorem 4.7 in [10]). Inthe representation V n ( a, b, c ), the central elements α, β and γ of ∆ q take the following values α = ( q n +1 + q − n − )( a + a − ) + ( b + b − )( c + c − ) , (5.3) β = ( q n +1 + q − n − )( b + b − ) + ( c + c − )( a + a − ) , (5.4) γ = ( q n +1 + q − n − )( c + c − ) + ( a + a − )( b + b − ) . (5.5)The characteristic polynomials of A, B, C ∈ ∆ q in this representation are (see Lemma 4.3 of [10]) K a ( X ) , K b ( X ) , K c ( X ), with(5.6) K x ( X ) = n Y i =0 ( X − ( q i − n x + q n − i x − )) . The Casimir element of the algebra ∆ q is given by(5.7) qAα + q − Bβ + qCγ − q A − q − B − q C − qABC . It is straightforward to compute the value ω of this element in the representation V n ( a, b, c ) by usingthe representation matrices given in [10]. One gets ω = ( q n +1 + q − n − ) + ( a + a − ) + ( b + b − ) + ( c + c − ) + ( q n +1 + q − n − )( a + a − )( b + b − )( c + c − ) − ( q + q − ) . (5.8)We want to find all the irreducible representations of ∆ q that pass to the quotient AW ( j , j , j ).By comparing the annihilating polynomials of the elements A, B, D ∈ AW ( j , j , j ) given in (4.13)–(4.14) with the characteristic polynomials (5.6), we get the following restrictions for the inequivalent representations V n ( a, b, c ) to pass to the quotient :0 ≤ n ≤ min { j + j − | j − j | , j + j − | j − j | , j + j − | j − j |} , (5.9) a = q x + n +1 , b = q y + n +1 , c = q z + n +1 , for x, y, z integers or half-integers,(5.10) | j − j | ≤ x ≤ j + j − n , | j − j | ≤ y ≤ j + j − n , | j − j | ≤ z ≤ j + j − n . (5.11)We recall that K is a central element with the annihilating polynomial given in (4.13). Therefore, K has to be a constant equal to χ ℓ for some ℓ ∈ J ( j , j , j ) in any irreducible representation of AW ( j , j , j ). We also recall that the Casimir element Ω of AW (3) satisfies relation (4.17) inthe quotient AW ( j , j , j ). From this discussion and from the results (5.2)–(5.5), (5.7), (5.8) and(5.10), we deduce that the following equations must hold so that the representation V n ( a, b, c ) passesto the quotient AW ( j , j , j ) : χ n χ x + n + χ y + n χ z + n = χ j χ j + χ j χ ℓ , (5.12) χ n χ y + n + χ z + n χ x + n = χ j χ j + χ j χ ℓ , (5.13) χ n χ z + n + χ x + n χ y + n = χ j χ j + χ j χ ℓ , (5.14) χ n + χ x + n + χ y + n + χ z + n + χ n χ x + n χ y + n χ z + n = χ j + χ j + χ j + χ ℓ + χ j χ j χ j χ ℓ . (5.15)In the case of three identical spins j = j = j = s , we find by using mathematical softwarethat there are only 192 possible solutions for x, y, z, n to the system of equations (5.12)–(5.15). Theonly solutions respecting conditions (5.9)–(5.11) and corresponding to inequivalent representations V n ( a, b, c ) are n = 2 ℓ , x = y = z = s − ℓ , if ℓ ≤ s , (5.16) n = 3 s − ℓ , x = y = z = ℓ − s , if ℓ > s , (5.17) n = s + ℓ , x = y = 0 , z = s − ℓ , if ℓ < s , (5.18) n = s − ℓ − , x = y = 0 , z = s + l + 1 , if ℓ ≤ s − , (5.19)and any permutation of x, y, z in the previous equations is also a solution.Since K = χ ℓ for ℓ ∈ J ( j , j , j ) in some irreducible representation, the annihilating polynomialof K − A given in (4.15) implies that the annihilating polynomial of A reduces to the relation (4.24)in this representation. If the set J ℓ ( j , j , j ) is equal to the set S ℓ ( j , j , j ), then this reducedannihilating polynomial for A leads to the constraint(5.20) max( | j − j | , | j − ℓ | ) ≤ x ≤ min( j + j , j + ℓ ) − n . Similar results hold for the annihilating polynomials of B (resp. D ) and the constraints on y (resp. z ). In the case of identical spins j = j = j = s , this implies that s − ℓ ≤ x, y, z ≤ s + ℓ − n for ℓ ≤ s , and ℓ − s ≤ x, y, z ≤ s − n for ℓ > s . The only solutions remaining are (5.16) and(5.17). For s half-integer, we do not find any cases where J ℓ ( s, s, s ) = S ℓ ( s, s, s ). For s integer, asa consequence of the fact that 0 ∈ M ( s, s, s ), we find J ℓ ( s, s, s ) \ S ℓ ( s, s, s ) = { ℓ } if ℓ < s/
2, andotherwise the previous set is empty. We have verified numerically the sets J ℓ ( s, s, s ) \ S ℓ ( s, s, s )given above for at least s = , , ...,
10. In any case, the upper bound on the values of x, y, z remainsthe same, and we still conclude that the only solutions are (5.16) and (5.17). Therefore, the sum of
SKEY–WILSON ALGEBRA AND CENTRALIZERS 15 the squares of the dimensions n + 1 of all the irreducible modules of AW ( s, s, s ) is(5.21) X ℓ ∈J ( s,s,s ) ℓ ≤ s (2 ℓ + 1) + X ℓ ∈J ( s,s,s ) ℓ>s (3 s − ℓ + 1) = 12 (2 s + 1)((2 s + 1) + 1) , which is equal to the dimension of the centralizer dim( C s,s,s ) = X ℓ ∈J ( s,s,s ) d ℓ .Let us remark that for j = j = j = , , ..., , we used mathematical software to test all thepossible integer values for x, y, z, n such that the restrictions (5.9) and (5.11) and the three equations(5.12)–(5.14) are respected. The only solutions we found are those given in (5.16)–(5.19). Hence,equation (5.15) is perhaps not necessary if one wants to find all solutions for x, y, z, n integers.Let us also notice that in the general case where j , j , j are any three fixed integers or half-integers, we find at least 192 solutions to the system of equations (5.12)–(5.14) with n integer and x, y, z integers or half-integers. If these are the only such solutions, then it is possible to argue (ina similar manner as for the case of identical spins) that the sum of the squares of the dimensions ofthe irreducible representations that pass to the quotient AW ( j , j , j ) is also equal to the dimensionof the centralizer C j ,j ,j .Finally, we notice that in the representations V n ( a, b, c ), the element D ′ of AW (3) has the samecharacteristic polynomial K c ( X ) as the element D . Therefore, the second relations in (4.14) and(4.16) do not provide any additional constraint on the values of n, a, b, c .6. Quotient AW ( , , ) and Temperley–Lieb algebra In this section, we consider the case j = j = j = and show that the quotient AW ( , , ) isisomorphic to the centralizer C , , , which is known to be the Temperley–Lieb algebra. We give anexplicit isomorphism between AW ( , , ) and the Temperley–Lieb algebra.6.1. AW ( , , ) algebra. From the definitions (3.2)–(3.3) and (3.14), we find the sets J (cid:18) , (cid:19) = { , } , J (cid:18) , , (cid:19) = (cid:26) , (cid:27) , (6.1) M (cid:18) , , (cid:19) = { χ − χ , χ − χ , χ − χ } . (6.2)The degeneracies are d = 2 and d = 1. We find from (3.8) that dim (cid:16) C , , (cid:17) = 5. The centralelements α i of AW ( , , ) can all be replaced by the constant χ . For computational convenience,we perform the transformation X = ( q − q − ) ˜ X + q + q − on the elements X = A, B, D, D ′ , K , asin Remark 2.1, and we define the shifted central element ˜ G = ˜ K + [1 / q . By using the sets given in (6.1) and (6.2), one finds that the defining relations of AW ( , , ) are[ ˜ B, [ ˜ A, ˜ B ] q ] q = [2] q (cid:16) − ˜ B − { ˜ A, ˜ B } (cid:17) + ( q + q − ) ˜ G ˜ B + [2] q ˜ B , (6.3) [[ ˜ A, ˜ B ] q , ˜ A ] q = [2] q (cid:16) − ˜ A − { ˜ A, ˜ B } (cid:17) + ( q + q − ) ˜ G ˜ A + [2] q ˜ A , (6.4) ˜ A ( ˜ A − [2] q ) = 0 , ˜ B ( ˜ B − [2] q ) = 0 , ( ˜ G − G − [2] q ) = 0 , (6.5) ˜ D ( ˜ D − [2] q ) = 0 , ˜ D ′ ( ˜ D ′ − [2] q ) = 0 , (6.6) ( ˜ G − ˜ A + [2] q − [2] q )( ˜ G − ˜ A + [2] q − G − ˜ A −
1) = 0 , (6.7) ( ˜ G − ˜ B + [2] q − [2] q )( ˜ G − ˜ B + [2] q − G − ˜ B −
1) = 0 , (6.8) ( ˜ G − ˜ D + [2] q − [2] q )( ˜ G − ˜ D + [2] q − G − ˜ D −
1) = 0 , (6.9) ( ˜ G − ˜ D ′ + [2] q − [2] q )( ˜ G − ˜ D ′ + [2] q − G − ˜ D ′ −
1) = 0 , (6.10) ( q + q − )( q − q − )( q ˜ A + q − ˜ B + q ˜ D ) ˜ G − [2] q (([2] q − q )( ˜ A + ˜ D ) + q − ˜ B )(6.11) − ( q − q − )( q ˜ A + q − ˜ B + q ˜ D ) − q [2] q ( q − q − )( ˜ A ˜ B + ˜ A ˜ D + ˜ B ˜ D ) − q ( q − q − ) ˜ A ˜ B ˜ D = ( q − q − ) ˜ G + ( q − q − − q [2] q ) ˜ G − q − [2] q , where ˜ D = [2] q + ( q + q − ) ˜ G − ˜ A − ˜ B − q − q − q + q − [ ˜ A, ˜ B ] q , (6.12) ˜ D ′ = [2] q + ( q + q − ) ˜ G − ˜ A − ˜ B − q − q − q + q − [ ˜ B, ˜ A ] q . (6.13)We want to show that AW ( , , ) is isomorphic to the centralizer C , , . Proposition 6.1.
The relations defining the quotient AW ( , , ) can be given as follows ˜ A = [2] q ˜ A , ˜ B = [2] q ˜ B , (6.14) ˜ A ˜ B ˜ A = [2] q { ˜ A, ˜ B } − [3] q ˜ A − [2] q ˜ B + [2] q [3] q , (6.15) ˜ B ˜ A ˜ B = [2] q { ˜ A, ˜ B } − [3] q ˜ B − [2] q ˜ A + [2] q [3] q . (6.16) Proof.
The two first relations in (6.5) directly lead to (6.14). The third relation in (6.5) implies(6.17) ˜ G = ([2] q + 1) ˜ G − [2] q . Developing (6.3) and (6.4) and using (6.14), one gets(6.18) ˜ B ˜ A ˜ B = ˜ G ˜ B , ˜ A ˜ B ˜ A = ˜ G ˜ A .
Expanding (6.7) and (6.8) and simplifying with the help of (6.14) and (6.17), one gets(6.19) ˜ G ˜ A = [2] q ˜ G + ˜ A − [2] q , ˜ G ˜ B = [2] q ˜ G + ˜ B − [2] q , which implies ˜ G ˜ A ˜ B = [2] q ˜ G + ˜ A ˜ B − [2] q , ˜ G ˜ B ˜ A = [2] q ˜ G + ˜ B ˜ A − [2] q . (6.20) SKEY–WILSON ALGEBRA AND CENTRALIZERS 17
Equations (6.6) and (6.9)–(6.11) can be simplified using the previous relations, and they lead to(6.21) ˜ G = − [2] q ( ˜ A + ˜ B ) + { ˜ A, ˜ B } + [2] q . Substituting (6.21) in (6.19) and (6.20), one finds˜ G ˜ A = [2] q { ˜ A, ˜ B } − [3] q ˜ A − [2] q ˜ B + [2] q [3] q , (6.22) ˜ G ˜ B = [2] q { ˜ A, ˜ B } − [3] q ˜ B − [2] q ˜ A + [2] q [3] q , (6.23) ˜ G ˜ A ˜ B = [2] q ˜ B ˜ A + ([2] q + 1) ˜ A ˜ B − [2] q ( ˜ A + ˜ B ) + [2] q [3] q , (6.24) ˜ G ˜ B ˜ A = [2] q ˜ A ˜ B + ([2] q + 1) ˜ B ˜ A − [2] q ( ˜ A + ˜ B ) + [2] q [3] q . (6.25)Equations (6.18) and (6.22)–(6.23) imply the relations (6.15)–(6.16) of the proposition.It remains to show that the generator ˜ G can be suppressed from the presentation, or in otherwords that (6.17) and (6.22)–(6.25) are implied from the relations of the proposition. Suppose thatrelations (6.14)–(6.16) are true and let ˜ G = − [2] q ( ˜ A + ˜ B ) + { ˜ A, ˜ B } + [2] q . Multiplying the expressionof ˜ G on the left and on the right by ˜ A and ˜ B , one finds(6.26) ˜ G ˜ A = ˜ A ˜ G = ˜ A ˜ B ˜ A , ˜ G ˜ B = ˜ B ˜ G = ˜ B ˜ A ˜ B .
Using (6.15) and (6.16), equations (6.22) and (6.23) are recovered. Multiplying (6.15) on the rightby ˜ B and (6.16) on the right by ˜ A , one finds˜ G ˜ A ˜ B = ˜ A ˜ B ˜ A ˜ B = ˜ A ˜ B − [2] q ˜ B + [2] q ˜ B ˜ A ˜ B , (6.27) ˜ G ˜ B ˜ A = ˜ B ˜ A ˜ B ˜ A = ˜ B ˜ A − [2] q ˜ A + [2] q ˜ A ˜ B ˜ A , (6.28)from which one easily recovers (6.24) and (6.25). Finally, it is straightforward to arrive at˜ G = − [2] q ( ˜ A + ˜ B ) + [2] q { ˜ A, ˜ B } − [2] q ( ˜ A ˜ B ˜ A + ˜ B ˜ A ˜ B ) + ˜ A ˜ B ˜ A ˜ B + ˜ B ˜ A ˜ B ˜ A + [2] q (6.29)and to use the results (6.27) and (6.28) to recover (6.17). (cid:3) Theorem 6.1.
Conjecture 4.1 is verified for j = j = j = .Proof. We already know from proposition 4.2 that the map φ is surjective. From the previousproposition, it is easy to show that { , ˜ A, ˜ B, ˜ A ˜ B, ˜ B ˜ A } is a linearly generating set of AW ( , , ).Since dim (cid:16) C , , (cid:17) = 5, this shows the injectivity of the map φ . (cid:3) Connection with the Temperley–Lieb algebra.
It is known that the Temperley–Liebalgebra is isomorphic to the centralizer of the diagonal embedding of U q ( sl ) in the tensor productof three fundamental representations [12]. Hence, from the results of the previous subsection, thequotiented Askey–Wilson algebra AW ( , , ) is isomorphic to the Temperley–Lieb algebra. Definition 6.1. [20]
The Temperley–Lieb algebra
T L ( q ) is generated by σ and σ with the follow-ing defining relations σ = ( q + q − ) σ , σ = ( q + q − ) σ , (6.30) σ σ σ = σ , σ σ σ = σ . (6.31) Theorem 6.2.
The quotiented Askey–Wilson algebra AW ( , , ) is isomorphic to the Temperley–Lieb algebra T L ( q ) . This isomorphism is given explicitly by AW (cid:18) , , (cid:19) → T L ( q )˜ A ( q + q − ) − σ , (6.32) ˜ B ( q + q − ) − σ . (6.33) Proof.
It is straightforward to show that the defining relations (6.30) and (6.31) of
T L ( q ) areequivalent to the relations (6.14)–(6.16) of AW ( , , ). (cid:3) Quotient AW (1 , , and Birman–Murakami–Wenzl algebra In this section, we choose j = j = j = 1 and prove that the quotient AW (1 , ,
1) is isomorphicto the centralizer C , , . In this case, C , , is known to be connected to the Birman–Murakami–Wenzl algebra. We give an explicit isomorphism between AW (1 , ,
1) and a specialization of theBMW algebra.7.1. AW (1 , , algebra. We have the following sets J (1 ,
1) = { , , } , J (1 , ,
1) = { , , , } , (7.1) M (1 , ,
1) = { χ − χ , χ − χ , , χ − χ , χ − χ , χ − χ } . (7.2)The degeneracies are d = d = 1, d = 3 and d = 2, and the dimension of the centralizer isdim( C , , ) = 15. The central elements α i of AW (1 , ,
1) can all be replaced by the constant χ .For computational convenience again, we perform the transformation X = ( q − q − ) ˜ X + q + q − on the elements X = A, B, D, D ′ , K (see Remark 2.1). We recall that the eigenvalues χ j aretransformed to ˜ χ j = [ j ] q [ j + 1] q , and we define the constants m = ˜ χ − ˜ χ , m = ˜ χ − ˜ χ and m = ˜ χ − ˜ χ . The defining relations (4.13)–(4.16) of AW (1 , ,
1) are written as( q + q − ) ˜ B ˜ A ˜ B = ˜ A ˜ B + ˜ B ˜ A − [2] q ˜ B − [2] q { ˜ A, ˜ B } + [2] q ([2] q −
3) ˜ K ˜ B + [2] q [3] q ˜ B , (7.3) ( q + q − ) ˜ A ˜ B ˜ A = ˜ B ˜ A + ˜ A ˜ B − [2] q ˜ A − [2] q { ˜ A, ˜ B } + [2] q ([2] q −
3) ˜ K ˜ A + [2] q [3] q ˜ A , (7.4) ˜ A ( ˜ A − ˜ χ )( ˜ A − ˜ χ ) = 0 , ˜ B ( ˜ B − ˜ χ )( ˜ B − ˜ χ ) = 0 , (7.5) ˜ K ( ˜ K − ˜ χ )( ˜ K − ˜ χ )( ˜ K − ˜ χ ) = 0 , (7.6) ˜ D ( ˜ D − ˜ χ )( ˜ D − ˜ χ ) = 0 , ˜ D ′ ( ˜ D ′ − ˜ χ )( ˜ D ′ − ˜ χ ) = 0 , (7.7) ( ˜ K − ˜ A + m )( ˜ K − ˜ A + m )( ˜ K − ˜ A )( ˜ K − ˜ A − m )( ˜ K − ˜ A − m )( ˜ K − ˜ A − m ) = 0 , (7.8) ( ˜ K − ˜ B + m )( ˜ K − ˜ B + m )( ˜ K − ˜ B )( ˜ K − ˜ B − m )( ˜ K − ˜ B − m )( ˜ K − ˜ B − m ) = 0 , (7.9) ( ˜ K − ˜ D + m )( ˜ K − ˜ D + m )( ˜ K − ˜ D )( ˜ K − ˜ D − m )( ˜ K − ˜ D − m )( ˜ K − ˜ D − m ) = 0 , (7.10) ( ˜ K − ˜ D ′ + m )( ˜ K − ˜ D ′ + m )( ˜ K − ˜ D ′ )( ˜ K − ˜ D ′ − m )( ˜ K − ˜ D ′ − m )( ˜ K − ˜ D ′ − m ) = 0 , (7.11) SKEY–WILSON ALGEBRA AND CENTRALIZERS 19 where ˜ D = [2] q [3] q + ([2] q −
3) ˜ K − ( q − q − )[2] q [ ˜ A, ˜ B ] q − ˜ A − ˜ B , (7.12) ˜ D ′ = [2] q [3] q + ([2] q −
3) ˜ K − ( q − q − )[2] q [ ˜ B, ˜ A ] q − ˜ A − ˜ B . (7.13)
Theorem 7.1.
Conjecture 4.1 is verified for j = j = j = 1 .Proof. We already know from proposition 4.2 that the map φ is surjective. We only need to provethat it is injective in this case.We define the set(7.14) S = { , ˜ A, ˜ B, ˜ A , ˜ A ˜ B, ˜ B ˜ A, ˜ B , ˜ A ˜ B, ˜ A ˜ B ˜ A, ˜ A ˜ B , ˜ B ˜ A , ˜ B ˜ A ˜ B, ˜ A ˜ B , ˜ A ˜ B ˜ A ˜ B, ˜ B ˜ A ˜ B ˜ A } . Using relations (7.3)–(7.6), it can be shown that S r = S ∪ ˜ K S ∪ ˜ K S ∪ ˜ K S is a linearly generatingset for AW (1 , , A r , ˜ B r and ˜ K r corresponding to theregular actions of ˜ A , ˜ B and ˜ K on the set S r . Knowing that ˜ A r , ˜ B r and ˜ K r have to satisfy (7.3)–(7.6)and the first relation of (7.7), we find 32 independant relations between the elements of S r and wecan reduce the generating set to(7.15) S ′ r = S ∪ ˜ K { , ˜ B, ˜ A , ˜ A ˜ B, ˜ B , ˜ A ˜ B, ˜ A ˜ B ˜ A, ˜ A ˜ B , ˜ B ˜ A ˜ B, ˜ A ˜ B , ˜ A ˜ B ˜ A ˜ B } ∪ ˜ K { ˜ A , ˜ A ˜ B } . We repeat the procedure and construct 28 by 28 matrices corresponding to the regular actions on S ′ r . Only using again (7.3)–(7.6) and the first of (7.7), we can reduce the generating set to(7.16) S ′′ r = S ∪ { ˜ K, ˜ K ˜ B, ˜ K ˜ A , ˜ K ˜ A ˜ B, ˜ K ˜ B , ˜ K ˜ A ˜ B, ˜ K ˜ A ˜ B ˜ A, ˜ K ˜ A ˜ B , ˜ K ˜ B ˜ A ˜ B } . We repeat and construct 24 by 24 matrices. At this point, relations (7.3)–(7.7) are already satisfied.We must use relations (7.8)–(7.10) to reduce the generating set to(7.17) S ′′′ r = S ∪ { ˜ K, ˜ K ˜ A , ˜ K ˜ B } . We repeat one last time by constructing 18 by 18 matrices and we use (7.3)–(7.11) to find 3independant relations which allow to reduce the generating set to S . It can also be verified thatthe matrices of the regular action satisfy the defining relation (4.17) involving the Casimir elementΩ. We made the previous computations by using a formal mathematical software.From these results, we have that S is a linearly generating set for AW (1 , ,
1) with 15 elements.Since dim ( C , , ) = 15, we conclude that φ is injective. (cid:3) Connection with the Birman–Murakami–Wenzl algebra.
It is known [16] that theBirman–Murakami–Wenzl algebra is isomorphic to the centralizer of the diagonal embedding of U q ( sl ) in the tensor product of three spin-1 representations. Hence, from the previous theorem,the quotiented Askey–Wilson algebra AW (1 , ,
1) is isomorphic to the BMW algebra.
Definition 7.1. [11]
The Birman–Murakami–Wenzl algebra
BM W ( Q, µ ) is generated by invertibleelements s and s with the following defining relations s s s = s s s , (7.18) e s = s e = µ − e , e s = s e = µ − e , (7.19) e s ǫ e = µ ǫ e , e s ǫ e = µ ǫ e , ǫ = ± , (7.20) e i = 1 − s i − s − i Q − Q − , i = 1 , . (7.21) Theorem 7.2.
The quotiented Askey–Wilson algebra AW (1 , , is isomorphic to the Birman–Murakami–Wenzl algebra BM W ( q , q ) . This isomorphism is given explicitly by AW (1 , , → BM W ( q , q )˜ A ( q + q − )( s − q − e ) + ( q + q − ) q − , (7.22) ˜ B ( q + q − )( s − q − e ) + ( q + q − ) q − . (7.23) Proof.
The algebras AW (1 , ,
1) and
BM W ( q , q ) are both isomorphic to C , , , hence they areisomorphic to each other. It can be verified that the image of ˜ A (resp. ˜ B ) in End( M ⊗ ) is equal tothe image of the R.H.S. of (7.22) (resp. (7.23)), which justifies the explicit mapping. The inversemap is given by s q − ( q + q − ) − ˜ A − q − ( q + q − ) − (2 + q − ) ˜ A + q − , (7.24) s q − ( q + q − ) − ˜ B − q − ( q + q − ) − (2 + q − ) ˜ B + q − . (7.25) (cid:3) Quotient AW ( , , )In this section, we take j = j = j = and show that the AW ( , , ) algebra is isomorphic tothe centralizer C , , .From the decomposition rules of the tensor product, we find the sets(8.1) J (cid:18) , (cid:19) = { , , , } , J (cid:18) , , (cid:19) = (cid:26) , , , , (cid:27) , M (cid:18) , , (cid:19) = { χ − χ , χ − χ , χ − χ , χ − χ , χ − χ , χ − χ ,χ − χ , χ − χ , χ − χ , χ − χ , χ − χ , χ − χ } . (8.2)The degeneracies are d = 1, d = d = 2, d = 3 and d = 4, and the dimension of the centralizeris dim (cid:16) C , , (cid:17) = 34. The central elements α i of AW ( , , ) are each equal to the constant χ .In order to prove the injectivity of the map φ in this case, we will use the strategy described inSubsection 4.3 and show that(8.3) dim (cid:18) K k AW (cid:18) , , (cid:19) K k (cid:19) ≤ d k ∀ k ∈ J (cid:18) , , (cid:19) . SKEY–WILSON ALGEBRA AND CENTRALIZERS 21
We recall that for each k ∈ J (cid:0) , , (cid:1) , the central element K is replaced by the constant χ k inthe algebra K k AW (cid:0) , , (cid:1) K k , and the annihilating polynomials for A (similarly for B , D and D ′ )reduce to(8.4) Y j ∈J k ( , , )( A − χ j ) = 0 , where J k (cid:0) , , (cid:1) = { j ∈ J (cid:0) , (cid:1) | χ j ∈ { χ k − m | m ∈ M (cid:0) , , (cid:1) }} . Once again, we performthe transformation X = ( q − q − ) ˜ X + q + q − on the elements X = A, B, D, D ′ , K (see Remark2.1). Therefore, one finds that the following relations hold in the algebras K k AW (cid:0) , , (cid:1) K k : • k = (8.5) ˜ A = ˜ B = χ . • k = ˜ B ˜ A ˜ B = [2] q { ˜ A, ˜ B } + [3] q ( − q ˜ A + ( q + q − −
1) ˜ B + [2] q ) , (8.6) ˜ A ˜ B ˜ A = [2] q { ˜ A, ˜ B } + [3] q ( − q ˜ B + ( q + q − −
1) ˜ A + [2] q ) , (8.7) ( ˜ A − ˜ χ )( ˜ A − ˜ χ ) = 0 , ( ˜ B − ˜ χ )( ˜ B − ˜ χ ) = 0 , (8.8) ( ˜ D − ˜ χ )( ˜ D − ˜ χ ) = 0 , ( ˜ D ′ − ˜ χ )( ˜ D ′ − ˜ χ ) = 0 . (8.9) • k = ( q + q − ) ˜ B ˜ A ˜ B = ˜ A ˜ B + ˜ B ˜ A − [2] q ˜ B − [2] q { ˜ A, ˜ B } + (2[2] q ˜ χ + ( q + q − )( ˜ χ + ˜ χ )) ˜ B , (8.10) ( q + q − ) ˜ A ˜ B ˜ A = ˜ B ˜ A + ˜ A ˜ B − [2] q ˜ A − [2] q { ˜ A, ˜ B } + (2[2] q ˜ χ + ( q + q − )( ˜ χ + ˜ χ )) ˜ A , (8.11) ( ˜ A − ˜ χ )( ˜ A − ˜ χ )( ˜ A − ˜ χ ) = 0 , ( ˜ B − ˜ χ )( ˜ B − ˜ χ )( ˜ B − ˜ χ ) = 0 , (8.12) ( ˜ D − ˜ χ )( ˜ D − ˜ χ )( ˜ D − ˜ χ ) = 0 , ( ˜ D ′ − ˜ χ )( ˜ D ′ − ˜ χ )( ˜ D ′ − ˜ χ ) = 0 . (8.13) • k = ( q + q − ) ˜ B ˜ A ˜ B = ˜ A ˜ B + ˜ B ˜ A − [2] q ˜ B − [2] q { ˜ A, ˜ B } + 2([2] q + q + q − ) ˜ χ ˜ B , (8.14) ( q + q − ) ˜ A ˜ B ˜ A = ˜ B ˜ A + ˜ A ˜ B − [2] q ˜ A − [2] q { ˜ A, ˜ B } + 2([2] q + q + q − ) ˜ χ ˜ A , (8.15) ˜ A ( ˜ A − ˜ χ )( ˜ A − ˜ χ )( ˜ A − ˜ χ ) = 0 , ˜ B ( ˜ B − ˜ χ )( ˜ B − ˜ χ )( ˜ B − ˜ χ ) = 0 , (8.16) ˜ D ( ˜ D − ˜ χ )( ˜ D − ˜ χ )( ˜ D − ˜ χ ) = 0 , ˜ D ′ ( ˜ D ′ − ˜ χ )( ˜ D ′ − ˜ χ )( ˜ D ′ − ˜ χ ) = 0 . (8.17) • k = ( q + q − ) ˜ B ˜ A ˜ B = [3] q ([2] q { ˜ A, ˜ B } − q ˜ A + [2] q ) + [2] q ( q + q − ) ˜ B , (8.18) ( q + q − ) ˜ A ˜ B ˜ A = [3] q ([2] q { ˜ A, ˜ B } − q ˜ B + [2] q ) + [2] q ( q + q − ) ˜ A , (8.19) ( ˜ A − ˜ χ )( ˜ A − ˜ χ ) = 0 , ( ˜ B − ˜ χ )( ˜ B − ˜ χ ) = 0 , (8.20) ( ˜ D − ˜ χ )( ˜ D − ˜ χ ) = 0 , ( ˜ D ′ − ˜ χ )( ˜ D ′ − ˜ χ ) = 0 . (8.21) For each value of k , the elements ˜ D and ˜ D ′ are given by˜ D = ( q + q − )[2] q ( ˜ χ + ˜ χ k ) + 2 ˜ χ − ( q − q − )[2] q [ ˜ A, ˜ B ] q − ˜ A − ˜ B , (8.22) ˜ D ′ = ( q + q − )[2] q ( ˜ χ + ˜ χ k ) + 2 ˜ χ − ( q − q − )[2] q [ ˜ B, ˜ A ] q − ˜ A − ˜ B . (8.23)
Theorem 8.1.
Conjecture 4.1 is verified for j = j = j = .Proof. Since we already know that the map φ is surjective, we only need to prove (8.3). For thecase k = , all the elements are constants and dim( K AW (cid:0) , , (cid:1) K ) = 1. For the case k = (resp. k = ), one uses (8.22) in the first relation of (8.9) (resp. (8.21)) to find (after somesimplifications using the defining relations ) ˜ B ˜ A = − ˜ A ˜ B + x ( ˜ A + ˜ B ) + x , for some constants x and x that can be computed. Therefore, in both cases we see that a linearly generating set isgiven by { , ˜ A, ˜ B, ˜ A ˜ B } . For the case k = , we used formal mathematical software to show that { , ˜ A, ˜ B, ˜ A , ˜ A ˜ B, ˜ B ˜ A, ˜ B , ˜ A ˜ B , ˜ B ˜ A } is a generating set. Similarly, for the case k = , a generatingset is given by { , ˜ A, ˜ B, ˜ A , ˜ A ˜ B, ˜ B ˜ A, ˜ B , ˜ A , ˜ A ˜ B ˜ A, ˜ A ˜ B , ˜ B ˜ A , ˜ B ˜ A ˜ B, ˜ B , ˜ A ˜ B, ˜ A ˜ B , ˜ A ˜ B } . Fromthese results and the degeneracies d k given at the beginning of the section, we see that (8.3) holds,which concludes the proof. (cid:3) Let us notice that the defining relation (4.17) of AW ( j , j , j ) which involves the Casimir elementΩ of AW (3) has not been called upon in the previous proof. It is straightforward to verify that thisrelation is satisfied in each of the algebras K k AW (cid:0) , , (cid:1) K k by using the relations given above.9. Quotient AW ( j, , ) and one-boundary Temperley–Lieb algebra In this section, we consider the case j = j , for j = 1 , , , ... , and j = j = . We show thatthe algebra AW ( j, , ) is isomorphic to the centralizer C j, , . We also find an explicit isomor-phism between this quotient of the Askey–Wilson algebra and a specialization of the one-boundaryTemperley–Lieb algebra.9.1. AW ( j, , ) algebra. From the tensor decomposition rules, we find the sets J (cid:18) j, (cid:19) = (cid:26) j − , j + 12 (cid:27) , J (cid:18) , (cid:19) = { , } , J (cid:18) j, , (cid:19) = { j − , j, j + 1 } , M (cid:18) j, , (cid:19) = { χ j +1 − χ j + , χ j − χ j + , χ j +1 − χ j − , χ j − − χ j − } ≡ { m , m , m , m } , M (cid:18) , , j (cid:19) = { χ j +1 − χ , χ j − χ , χ j − χ , χ j − − χ } . The degeneracies are d j − = d j +1 = 1 and d j = 2, and the dimension of the centralizer isdim( C j, , ) = 6. The central elements α i take the values α = χ j and α = α = χ in the quotient AW ( j, , ). As in the previous sections, we perform the transformation X = ( q − q − ) ˜ X + q + q − on the generators X = A, B, D, D ′ , K (see Remark 2.1). The defining relations of AW ( j, , ) can SKEY–WILSON ALGEBRA AND CENTRALIZERS 23 be written as follows˜ B ˜ A ˜ B = (cid:0) ˜ χ j − [2] q [1 / q (cid:1) ˜ B + ˜ K ˜ B , (9.1) ˜ A ˜ B ˜ A = a { ˜ A, ˜ B } − a ˜ B + a ˜ A + ˜ K ( ˜ A − a ) + a , (9.2) ( ˜ A − ˜ χ j − )( ˜ A − ˜ χ j + ) = 0 , ˜ B ( ˜ B − [2] q ) = 0 , ( ˜ K − ˜ χ j − )( ˜ K − ˜ χ j )( ˜ K − ˜ χ j +1 ) = 0 , (9.3) ( ˜ D − ˜ χ j − )( ˜ D − ˜ χ j + ) = 0 , ( ˜ D ′ − ˜ χ j − )( ˜ D ′ − χ j + ) = 0 , (9.4) ( ˜ K − ˜ B − ˜ χ j +1 + ˜ χ )( ˜ K − ˜ B − ˜ χ j + ˜ χ )( ˜ K − ˜ B − ˜ χ j )( ˜ K − ˜ B − ˜ χ j − + ˜ χ ) = 0 , (9.5) Y i =1 ( ˜ K − ˜ A − m i ) = 0 , Y i =1 ( ˜ K − ˜ D − m i ) = 0 , Y i =1 ( ˜ K − ˜ D ′ − m i ) = 0 , (9.6) Ω = χ j + 2 χ + K + χ j χ K − χ , (9.7)where ˜ D = ( q + q − )[2] q ( ˜ K + ˜ χ j ) + 2 ˜ χ − ( q − q − )[2] q [ ˜ A, ˜ B ] q − ˜ A − ˜ B , (9.8) ˜ D ′ = ( q + q − )[2] q ( ˜ K + ˜ χ j ) + 2 ˜ χ − ( q − q − )[2] q [ ˜ B, ˜ A ] q − ˜ A − ˜ B , (9.9) Ω = q ( A + D ) χ ( χ j + K ) + q − B ( χ + χ j K ) − q A − q − B − q D − qABD , (9.10)and where we have used the following constants a = [2] q [ j − ] q [ j + ] q q + q − , a = 2 ˜ χ j − ˜ χ j + q + q − ,a = [2] q q + q − (2 ˜ χ − [2] q [ j + ] q ) + ˜ χ j , a = a ([ j + ] q + ˜ χ ) ,m = ˜ χ j +1 − ˜ χ j + , m = ˜ χ j − ˜ χ j + , m = ˜ χ j +1 − ˜ χ j − , m = ˜ χ j − − ˜ χ j − . Proposition 9.1.
The quotient AW ( j, , ) can be presented with the following relations ˜ A = ( ˜ χ j − + ˜ χ j + ) ˜ A − ˜ χ j − ˜ χ j + , ˜ B = [2] q ˜ B , (9.11) ˜ B ˜ A ˜ B = [2] q { ˜ A, ˜ B } − [2] q ˜ A − ([ j + ] q + [ j − ] q − B − [2] q ) . (9.12) Proof.
We first show that the relations (9.1)–(9.6) imply the relations of the proposition. The twoequations in (9.11) follow directly from (9.3). We also deduce from the first relation of (9.3) that(9.13) (cid:16) ˜ A − a (cid:17) (cid:16) ˜ A − ˜ χ j − − ˜ χ j + + a (cid:17) = [2 j − q [2 j + 3] q q + q − . Since the R.H.S. of the previous relation does not vanish for j > , it can be used in (9.2) to find(9.14) ˜ K = { ˜ A, ˜ B } − ( ˜ χ j − + ˜ χ j + ) ˜ B − [2] q ˜ A + (1 + [2] q )([2 j + ] q [ ] q + ˜ χ j − ) . Using (9.14) in (9.8) and (9.9), and then substituting in (9.4), one obtains expressions for ˜ A ˜ B ˜ A ˜ B and ˜ B ˜ A ˜ B ˜ A in terms of the elements 1 , ˜ A, ˜ B, ˜ A ˜ B, ˜ B ˜ A, ˜ A ˜ B ˜ A and ˜ B ˜ A ˜ B . By using (9.14) and theexpressions for ˜ A ˜ B ˜ A ˜ B and ˜ B ˜ A ˜ B ˜ A in the third relation of (9.3), one gets (9.12). Finally, we want to show that (9.11) and (9.12) imply the defining relations of AW ( j, , ). Tothat end, we suppose that the relations of the proposition are true and we define the element ˜ K asin (9.14). It is then straightforward to verify that ˜ K is central and that (9.1)–(9.7) hold. (cid:3) Theorem 9.1.
Conjecture 4.1 is verified for j = j and j = j = , where j = 1 , , , ... Proof.
We already know that the map φ is surjective. From the previous proposition, we concludethat { , ˜ A, ˜ B, ˜ A ˜ B, ˜ B ˜ A, ˜ A ˜ B ˜ A } is a generating set for AW ( j, , ). Therefore, dim (cid:0) AW ( j, , ) (cid:1) ≤ dim (cid:16) C j, , (cid:17) = 6, which shows the injectivity of φ . (cid:3) Connection with the one-boundary Temperley–Lieb algebra.
On the basis of the find-ings for the limit q → j and twospin- will be isomorphic to the one-boundary Temperley–Lieb algebra. We can indeed confirmthat this algebra is recovered as a quotient of AW (3). Definition 9.1. [17, 18, 19]
The one-boundary Temperley–Lieb algebra bT L ( q, ω ) is generated by σ and σ with the following defining relations (9.15) σ = [ ω ] q [ ω − q σ , σ = ( q + q − ) σ , σ σ σ = σ . Theorem 9.2.
The quotiented Askey–Wilson algebra AW ( j, , ) , for j = 1 , , ... , is isomorphicto the one-boundary Temperley–Lieb algebra bT L ( q, j + 1) . This isomorphism is given explicitlyby AW (cid:18) j, , (cid:19) → bT L ( q, j + 1)˜ A ˜ χ j + − [2 j ] q σ , (9.16) ˜ B [2] q − σ . (9.17) Proof.
It is easy to see that the map ϕ is bijective. The homomorphism can be directly verifiedfrom the relations of the proposition 9.1. (cid:3) Conclusion and perspectives
Summing up, we have offered a conjecture according to which a quotient of the Askey–Wilsonalgebra is isomorphic to the centralizer of the image of the diagonal embedding of U q ( sl ) in thetensor product of any three irreducible representations. It has been proved in several cases, and wethus obtained the Temperley–Lieb, Birman–Murakami–Wenzl and one-boundary Temperley–Liebalgebras as quotients of the Askey–Wilson algebra. In the limit q →
1, the results of the paper[4] are recovered. We have provided further evidence in support of the conjecture by studying thefinite irreducible representations of the quotient of the Askey–Wilson algebra, more particularly inthe case of three identical spins.Proving the conjecture in the case of three arbitrary spins j , j , j would be an obvious con-tinuation of the work presented here. If true, this conjecture would provide a presentation of thecentralizer of U q ( sl ) in terms of generators and relations for any three irreducible representations. SKEY–WILSON ALGEBRA AND CENTRALIZERS 25
We could first consider, more simply, the case of three identical spins j = j = j = s . As forthe Temperley–Lieb ( s = ) and the Birman–Murakami–Wenzl ( s = 1) algebras, we expect thatthe centralizer for any spin s will be linked to a quotient of the braid group algebra.In [3], a diagrammatic description of the centralizers of U q ( gl n ) has been proposed. It is based onthe notion of fused Hecke algebras. Developing a connection between this diagrammatic approachand the Askey–Wilson algebra could prove fruitful.Throughout the present paper, we assume q to be not a root of unity. This choice allows todecompose the tensor product of irreducible representations of U q ( sl ) into a direct sum of irreduciblerepresentations (see Subsection 3.1). As a consequence, the matrices C i , C ij , C are diagonalizableand their minimal polynomials are those discussed in Subsection 3.2. It could be interesting to studythe centralizer when q is a root of unity and to examine how the quotient of AW (3) is affected.Another generalization of the results presented here would be to consider the n -fold tensor productof irreducible representations of U q ( sl ) and to connect the centralizer to a higher rank Askey–Wilsonalgebra AW ( n ). The approach using the R -matrix proposed in [2] should be helpful for this purpose.In fact, a simpler starting point could be to generalize either the conjecture given in [4] by studyingthe connection between the centralizers of sl and a higher rank Racah algebra, or the one givenin [1] by examining how centralizers of osp (1 |
2) relate to the higher rank Bannai-Ito algebra BI ( n )(see [6]).Yet another direction to generalize the results of this paper would be to study the centralizerof the diagonal embedding of g or U q ( g ) with g a higher rank Lie algebra. A first step in thisdirection was made recently in [5] where the centralizer Z ( sl ) of the diagonal embedding of sl inthe twofold tensor product of sl has been identified. A proposition similar to Proposition 4.2 hasalso been proved in that case. A quotient of the algebra Z ( sl ) that describes the centralizer forany representations of sl has still to be investigated. We hope to report on some of this issues inthe future. Acknowledgments:
The authors are grateful to Lo¨ıc Poulain D’Andecy for numerous enlight-ening discussions. They also thank Paul Terwilliger for useful exchanges. N. Cramp´e is partiallysupported by Agence Nationale de la Recherche Projet AHA ANR-18-CE40-0001. The work ofL. Vinet is funded in part by a discovery grant of the Natural Sciences and Engineering ResearchCouncil (NSERC) of Canada. M. Zaimi holds graduate scholarships from the NSERC and theFonds de recherche du Qu´ebec - Nature et technologies (FRQNT).
References [1] N. Crampe, L. Frappat and L. Vinet,
Centralizers of the superalgebra osp (1 | : the Brauer algebra as a quotientof the Bannai–Ito algebra, J. Phys. A: Math. Theor. 52 (2019) 424001 and arXiv:1906.03936 .[2] N. Crampe, J. Gaboriaud, L. Vinet and M. Zaimi,
Revisiting the Askey–Wilson algebra with the universal R -matrix of U q ( sl (2)) , J. Phys. A: Math. Theor. 53 (2020) 05LT01 and arXiv:1908.04806v2 .[3] N. Crampe and L. Poulain d’Andecy,
Fused braids and centralisers of tensor representations of U q ( gl N ) , (2020) arXiv:2001.11372 .[4] N. Crampe, L. Poulain d’Andecy and L. Vinet, Temperley–Lieb, Brauer and Racah algebras and other central-izers of su (2) , Trans. Amer. Math. Soc. 373 (2020) 4907–4932 and arXiv:1905.06346 .[5] N. Crampe, L. Poulain d’Andecy and L. Vinet,
A Calabi-Yau algebra with E symmetry and the Clebsch-Gordanseries of sl (3) , (2020) arXiv:2005.13444 .[6] N. Crampe, L. Vinet and M. Zaimi, Bannai–Ito algebras and the universal R -matrix of osp (1 | , Lett. Math.Phys. 110 (2020) 1043–1055 and arXiv:1909.06426 . [7] V.G. Drinfeld, Quantum groups, in: Proc. ICM (Berkeley,1986), Vol.1 (Academic Press, New York, 1987)pp.798–820.[8] Ya.A. Granovskii and A.S. Zhedanov,
Hidden symmetry of the racah and Clebsch-Gordan problems for thequantum algebra sl q (2) , Journal of Group Theory in Physics 1 (1993) 161–171 and arXiv:hep-th/9304138 .[9] H.-W. Huang,
An embedding of the universal Askey–Wilson algebra into U q ( sl ) ⊗ U q ( sl ) ⊗ U q ( sl ) , Nucl. Phys.B, 922 (2017) 401–434 and arXiv:1611.02130 .[10] H.-W. Huang,
Finite-dimensional irreducible modules of the universal Askey–Wilson algebra,
Comm. in Math.Phys., (2015) and arXiv:1210.1740 .[11] A. P. Isaev, A. I. Molev and O. V. Ogievetsky,
Idempotents for Birman–Murakami–Wenzl algebras and reflectionequation,
Adv. Theor. Math. Phys. (2011) and arXiv:1111.2502 .[12] M. Jimbo,
A q-Analogue of U ( gl ( N + 1)) , Hecke Algebra, and the Yang–Baxter Equation, Lett. Math. Phys.11 (1986) 247–252.[13] R. Koekoek, P.A. Lesky and R.F. Swarttouw,
Hypergeometric orthogonal polynomials and their q -analogues ,Springer, 1-st edition (2010).[14] H. Kraft and C. Procesi, Classical invariant theory : a primer , (1996).[15] G.I. Lehrer and R.B. Zhang,
Strongly multiplicity free modules for Lie algebras and quantum groups , Journalof Algebra 306 (2006) 138–174.[16] G.I. Lehrer and R.B. Zhang,
A Temperley–Lieb analogue for the BMW algebra , in Representation theory ofalgebraic groups and quantum groups. Birkhuser Boston, (2010) 155–190 and arXiv:0806.0687v1 .[17] P.P. Martin and H. Saleur,
On an algebraic approach to higher dimensional statistical mechanics , Commun.Math. Phys. 158 (1993) 155–190 and arXiv:hep-th/9208061 .[18] P.P. Martin and D. Woodcock,
On the structure of the blob algebra , Journal of Algebra 225 (2000) 957–988.[19] A. Nichols, V. Rittenberg and J. de Gier,
One-boundary Temperley–Lieb algebras in the XXZ and loop models ,J. Stat. Mech. 0503 (2005) P03003 and arXiv:cond-mat/0411512 .[20] N. Temperley and E. Lieb,
Relations between the ’Percolation’ and ’Colouring’ Problem and other Graph-Theoretical Problems Associated with Regular Planar Lattices: Some Exact Results for the ’Percolation’ Prob-lem.
Proc. Royal Soc. A 322 (1971) 251–280.[21] P. Terwilliger,
The Universal Askey–Wilson Algebra,
SIGMA 7 (2011), 069 and arXiv:1104.2813 .[22] A.S. Zhedanov,
Hidden symmetry of the Askey–Wilson polynomials,
Theor. Math. Phys. 89 (1991) 1146–1157. † Institut Denis-Poisson CNRS/UMR 7013 - Universit´e de Tours - Universit´e d’Orl´eans, Parc deGrandmont, 37200 Tours, France.
E-mail address : [email protected] ∗ Centre de recherches math´ematiques, Universit´e de Montr´eal, P.O. Box 6128, Centre-villeStation, Montr´eal (Qu´ebec), H3C 3J7, Canada.
E-mail address : [email protected] ∗∗ Centre de recherches math´ematiques, Universit´e de Montr´eal, P.O. Box 6128, Centre-villeStation, Montr´eal (Qu´ebec), H3C 3J7, Canada.
E-mail address ::