Higher Rank Relations for the Askey-Wilson and q -Bannai-Ito Algebra
SSymmetry, Integrability and Geometry: Methods and Applications SIGMA (2019), 099, 32 pages Higher Rank Relations for the Askey–Wilsonand q -Bannai–Ito Algebra Hadewijch DE CLERCQDepartment of Electronics and Information Systems, Faculty of Engineering and Architecture,Ghent University, Belgium
E-mail: [email protected]
Received September 03, 2019, in final form December 13, 2019; Published online December 19, 2019https://doi.org/10.3842/SIGMA.2019.099
Abstract.
The higher rank Askey–Wilson algebra was recently constructed in the n -foldtensor product of U q ( sl ). In this paper we prove a class of identities inside this algebra,which generalize the defining relations of the rank one Askey–Wilson algebra. We extendthe known construction algorithm by several equivalent methods, using a novel coaction.These allow to simplify calculations significantly. At the same time, this provides a proof ofthe corresponding relations for the higher rank q -Bannai–Ito algebra. Key words:
Askey–Wilson algebra; Bannai–Ito algebra
The Askey–Wilson algebra was introduced in [33] as an algebraic foundation for the bispectralproblem of the Askey–Wilson orthogonal polynomials [22]. More precisely, Zhedanov definedthis algebra by generators and relations, which turned out to be satisfied when realizing thegenerators as the Askey–Wilson q -difference operator on the one hand, and the multiplicationwith the variable on the other. A central extension, which allows a Z -symmetric presentation,was defined and studied by Terwilliger [29]. He calls this central extension the universal Askey–Wilson algebra. We will denote it by AW(3).The irreducible finite-dimensional representations of the Askey–Wilson algebra have beenclassified by Leonard pairs [26, 31] and a similar classification for the universal Askey–Wilsonalgebra AW(3) has appeared in [18]. Further applications arise in the theory of special func-tions [4, 24], tridiagonal and Leonard pairs [25, 28, 32], superintegrable quantum systems [3] andthe reflection equation [2]. Its use to quantum mechanics is further emphasized by the identifi-cation of the Askey–Wilson algebra as a quotient of the q -Onsager algebra [27], which originatesfrom statistical mechanics. This connection was later extended to the universal Askey–Wilsonalgebra [30]. Furthermore, an explicit homomorphism from the original Askey–Wilson algebrato the double affine Hecke algebra of type (cid:0) C ∨ , C (cid:1) has been constructed in [21, 23]. Recently,connections with q -Higgs algebras [13] and Howe dual pairs [14] have been obtained.The original Askey–Wilson algebra [33] was realized inside the quantum group U q ( sl ) in [17].A different realization in the three-fold tensor product of U q ( sl ) was given in [16], and was laterextended to the universal Askey–Wilson algebra AW(3) of [29] in [19]. In the latter reference,AW(3) is embedded in the threefold tensor product of U q ( sl ): if one denotes by Λ the quadraticCasimir element of U q ( sl ) and by ∆ its coproduct, and definesΛ { } = Λ ⊗ ⊗ , Λ { } = 1 ⊗ Λ ⊗ , Λ { } = 1 ⊗ ⊗ Λ , Λ { , } = ∆(Λ) ⊗ , Λ { , } = 1 ⊗ ∆(Λ) , Λ { , , } = (1 ⊗ ∆)∆(Λ) , (1.1) a r X i v : . [ m a t h . QA ] D ec H. De Clercqthen these elements generate the universal Askey–Wilson algebra. Indeed, they satisfy the q -commutation relations[Λ { , } , Λ { , } ] q = (cid:0) q − − q (cid:1) Λ { , } + (cid:0) q − q − (cid:1)(cid:0) Λ { } Λ { , , } + Λ { } Λ { } (cid:1) , (1.2)[Λ { , } , Λ { , } ] q = (cid:0) q − − q (cid:1) Λ { , } + (cid:0) q − q − (cid:1)(cid:0) Λ { } Λ { , , } + Λ { } Λ { } (cid:1) , (1.3)[Λ { , } , Λ { , } ] q = (cid:0) q − − q (cid:1) Λ { , } + (cid:0) q − q − (cid:1)(cid:0) Λ { } Λ { , , } + Λ { } Λ { } (cid:1) , (1.4)where Λ { , } is defined through relation (1.2), and Λ { , , } and all Λ { i } are central. This coincideswith the presentation for AW(3) given in [29]. This illustrates that there exists an algebrahomomorphism from AW(3) to U q ( sl ) ⊗ and this map turns out to be injective, as shown in[19, Theorem 4.8].In [8] this approach was generalized to n -fold tensor products for arbitrary n , which leadsto an extension of the universal Askey–Wilson algebra to higher rank, denoted AW( n ). Thesame algorithm allows to construct a higher rank extension of the q -Bannai–Ito algebra, whichis isomorphic to AW(3) under a transformation q → − q and allows a similar embedding in osp q (1 | ⊗ [15]. Also the limiting cases q = 1 provide interesting algebras, as summarizedgraphically below. Askey–WilsonalgebraRacah algebra q -Bannai–ItoalgebraBannai–Itoalgebra q → q → − q q → Such higher rank algebras are motivated by their role as symmetry algebras for superinte-grable quantum systems of higher dimension. This has been confirmed in the limiting case q = 1[9, 10, 11], and later also for general q [8]. In both cases, the Hamiltonians under considerationare built from Dunkl operators with Z n symmetry [12], possibly q -deformed [5]. Moreover, thesehigher rank algebras allow to extend known connections with orthogonal polynomials to multiplevariables. This was achieved in [7] for the q -Bannai–Ito algebra. To be concrete, an action of thehigher rank q -Bannai–Ito algebra on an abstract vector space was considered, leading to variousorthonormal bases for this space. The overlap coefficients between such bases turned out tobe multivariable ( − q )-Racah polynomials, the truncated analogs of Askey–Wilson polynomials.This has allowed to construct a realization of the higher rank q -Bannai–Ito algebra with Iliev’s q -difference operators [20], which have thereby obtained an algebraic interpretation.The construction of AW( n ) is rather intricate: in [8] we have outlined an algorithm whichrepeatedly applies the U q ( sl )-coproduct ∆ and a coaction τ to the Casimir element Λ, ina specific order. This way we construct, as an extension of (1.2)–(1.4), an element Λ A ∈ U q ( sl ) ⊗ n for each A ⊆ { , . . . , n } . In this paper we rephrase this extension algorithm in more accessiblenotation and provide alternative construction methods for the elements Λ A which use a novelcoaction. This new approach is of major use to derive algebraic identities in AW( n ), as weshowcase in Theorems 3.1 and 3.2 by significantly generalizing the algebraic relations givenin [8]. The main achievement of this paper is hence a general criterion for two generators Λ A and Λ B of AW( n ) to commute or to satisfy a relation of the form[Λ A , Λ B ] q = (cid:0) q − − q (cid:1) Λ ( A ∪ B ) \ ( A ∩ B ) + (cid:0) q − q − (cid:1)(cid:0) Λ A ∩ B Λ A ∪ B + Λ A \ ( A ∩ B ) Λ B \ ( A ∩ B ) (cid:1) . (1.5)More precisely, we show in Theorem 3.1 that[Λ A , Λ B ] = 0 if B ⊆ A, igher Rank Relations for the Askey–Wilson and q -Bannai–Ito Algebra 3and in Theorem 3.2 we prove that the relation (1.5) is satisfied for A = A ∪ A ∪ A , B = A ∪ A ,A = A ∪ A , B = A ∪ A ∪ A ,A = A ∪ A ∪ A , B = A ∪ A ∪ A , (1.6)where A , A , A , A ⊆ { , . . . , n } are such that for each i ∈ { , , } one has either max( A i ) < min( A i +1 ) or A i = ∅ or A i +1 = ∅ .It is not clear at this point whether these relations define the algebra AW( n ) abstractly, or,in case the answer turns to be negative, which supplementary relations should be added in orderto attain this purpose. However, calculations with computer algebra packages suggest that thecondition (1.6) describes the most general situation for the relations (1.5) to be satisfied.Our methods are elementary and intrinsic: they are independent of the expressions for thecoactions and the U q ( sl )-Casimir Λ, and only recur to natural algebraic properties like coas-sociativity and the cotensor product property. As a consequence, the results of this paper areequally applicable to the higher rank q -Bannai–Ito algebra of [8], without any modification.The paper is organized as follows. In Section 2 we construct the higher rank Askey–Wilsonalgebra AW( n ) as a subalgebra of U q ( sl ) ⊗ n through different extension processes, which weprove to be equivalent. Section 3 lists the main results of this paper and explains the generalstrategy of proof. Consequently, in Sections 4 and 5 we prove some intermediate results whichwill be relied on in Sections 6 and 7, where we prove Theorems 3.1 and 3.2. Finally in Section 8we introduce similar extension processes to construct a higher rank q -Bannai–Ito algebra asa subalgebra of osp q (1 | ⊗ n . We state two explicit theorems describing their algebraic relations. Throughout this paper, we will work with the quantum group U q ( sl ), which can be presentedas the associative algebra over a field K with generators E , F , K and K − , and relations KK − = K − K = 1 , KE = q EK, KF = q − F K, [ E, F ] = K − K − q − q − , where q is a fixed parameter in K , assumed not to be a root of unity. A Casimir element, whichcommutes with all elements of U q ( sl ), is given byΛ = (cid:0) q − q − (cid:1) EF + q − K + qK − . (2.1)The quantum group U q ( sl ) has the structure of a bialgebra: it is equipped with a coproduct∆ : U q ( sl ) → U q ( sl ) ⊗ , which satisfies the coassociativity property (1 ⊗ ∆)∆ = (∆ ⊗ (cid:15) : U q ( sl ) → K satisfying (1 ⊗ (cid:15) )∆ = ( (cid:15) ⊗ U q ( sl ). Explicitly, they are given by∆( E ) = E ⊗ K ⊗ E, ∆( F ) = F ⊗ K − + 1 ⊗ F, ∆ (cid:0) K ± (cid:1) = K ± ⊗ K ± , (2.2) (cid:15) ( E ) = (cid:15) ( F ) = 0 , (cid:15) ( K ) = (cid:15) (cid:0) K − (cid:1) = 1 . A binary operation we will often use is the so-called q -commutator. For X, Y ∈ U q ( sl ) wewrite [ X, Y ] q = qXY − q − Y X.
For i and j natural numbers with i ≤ j , we will write [ i ; j ] to denote the discrete interval { i, i + 1 , . . . , j − , j } . If we consider disjoint unions of discrete intervals, often denoted by[ i ; j ] ∪ [ i ; j ] ∪ · · · ∪ [ i k ; j k ], it is always understood that i (cid:96) ≤ j (cid:96) < i (cid:96) +1 − (cid:96) . Note thatthis implies that i k ≥ i + 2 k −
2. Moreover, if B is any set of natural numbers and a ∈ N , wewill write B − a for the set { b − a : b ∈ B } . H. De Clercq In [8] we have introduced the following U q ( sl )-subalgebra. Definition 2.1.
We denote by I R the subalgebra of U q ( sl ) generated by EK − , F , K − and Λ,and we define the algebra morphism τ R : I R → U q ( sl ) ⊗ I R through its action on the generators: τ R (cid:0) EK − (cid:1) = K − ⊗ EK − ,τ R ( F ) = K ⊗ F − q − (cid:0) q − q − (cid:1) F K ⊗ EK − + q − (cid:0) q + q − (cid:1) F K ⊗ K − − q − F K ⊗ Λ ,τ R (cid:0) K − (cid:1) = 1 ⊗ K − − q − (cid:0) q − q − (cid:1) F ⊗ EK − ,τ R (Λ) = 1 ⊗ Λ . It is readily checked that these definitions comply with the algebra relations in I R . Thefollowing important observation was made in [8, Proposition 3]. Proposition 2.1.
The algebra I R is a left coideal subalgebra of U q ( sl ) and a left U q ( sl ) -comodule with coaction τ R . This means that ∆( I R ) ⊂ U q ( sl ) ⊗ I R and that one has (1 ⊗ τ R ) τ R = (∆ ⊗ τ R , (2.3)( (cid:15) ⊗ τ R = 1 . An interpretation of the coaction τ R in terms of the universal R -matrix for U q ( sl ) wasrecently given in [6]. This coaction was constructed so as to satisfy the identityΛ { , } = (1 ⊗ τ R )∆(Λ) , (2.4)with Λ { , } defined through (1.1) and (1.2). A similar mapping τ L can be constructed bydemanding thatΛ { , } = ( τ L ⊗ . (2.5)This suggests the following definition. Definition 2.2.
We denote by I L the subalgebra of U q ( sl ) generated by E , F K , K and Λ, andwe define the algebra morphism τ L : I L → I L ⊗ U q ( sl ) through its action on the generators: τ L ( E ) = E ⊗ K,τ L ( F K ) =
F K ⊗ K − − q − (cid:0) q − q − (cid:1) E ⊗ F K + q (cid:0) q + q − (cid:1) K ⊗ F − q Λ ⊗ F,τ L ( K ) = K ⊗ − q − (cid:0) q − q − (cid:1) E ⊗ F K,τ L (Λ) = Λ ⊗ . This subalgebra behaves in a similar fashion.
Proposition 2.2.
The algebra I L is a right coideal subalgebra of U q ( sl ) and a right U q ( sl ) -comodule with coaction τ L . Proof .
It suffices to check explicitly on each of the generators that1) I L is a right coideal of U q ( sl ): ∆( I L ) ⊂ I L ⊗ U q ( sl ),2) τ L is a right coaction: it preserves the algebra relations in I L and satisfies( τ L ⊗ τ L = (1 ⊗ ∆) τ L , (2.6)(1 ⊗ (cid:15) ) τ L = 1 . (cid:4) igher Rank Relations for the Askey–Wilson and q -Bannai–Ito Algebra 5It is readily checked that the element ∆(Λ) lies in I L ⊗ I R . Indeed, it follows immediatelyfrom (2.1) and (2.2) that one has∆(Λ) = Λ ⊗ K − + K ⊗ Λ − (cid:0) q + q − (cid:1) K ⊗ K − + (cid:0) q − q − (cid:1) (cid:0) E ⊗ F + q − F K ⊗ EK − (cid:1) . In the language of category theory [1], the equality of (2.4) and (2.5) can be phrased asfollows.
Corollary 2.1.
The element ∆(Λ) ∈ I L ⊗ I R , with Λ defined in (2.1) , belongs to the cotensorproduct of the coideal comodule subalgebras I L and I R of U q ( sl ) : (1 ⊗ τ R )∆(Λ) = ( τ L ⊗ . (2.7) Our goal in this section will be to associate to each set A ⊆ [1; n ] an element Λ A ∈ U q ( sl ) ⊗ n ,which will serve as a generator for the higher rank Askey–Wilson algebra AW( n ). For the emptyset, this will simply be the scalarΛ ∅ = q + q − . (2.8)For general A , a construction algorithm was given in [8]. We will repeat it here in a moreaccessible notation. Definition 2.3.
For any set A = { a , . . . , a m } ⊆ [1; n ], ordered such that a i < a i +1 for all i , wedefine Λ A ∈ U q ( sl ) ⊗ n byΛ A = 1 ⊗ ( a − ⊗ −→ m (cid:89) i =2 µ Ai (Λ) ⊗ ⊗ ( n − a m ) , with µ Ai = −−−−−−−→ a i − a − (cid:89) (cid:96) = a i − − a +1 (cid:0) ⊗ (cid:96) ⊗ τ R (cid:1) (cid:0) ⊗ ( a i − − a ) ⊗ ∆ (cid:1) , (2.9)where it is understood that the term between brackets in (2.9) is absent if a i = a i − + 1. Example 2.1. If n = 9 and A = { , , , } , then we haveΛ { , , , } = 1 ⊗ (cid:0) µ { , , , } µ { , , , } µ { , , , } (cid:1) (Λ) ⊗ , with µ { , , , } = (1 ⊗ τ R )∆, µ { , , , } = 1 ⊗ ⊗ ∆ and µ { , , , } = (cid:0) ⊗ ⊗ τ R (cid:1)(cid:0) ⊗ ⊗ τ R (cid:1)(cid:0) ⊗ ⊗ ∆ (cid:1) .The rationale behind this construction is that each element of A , except for its minimum a ,corresponds to an application of ∆, whereas the coaction τ R is used to create the gaps betweenthe elements of A . In the example above, τ R is applied first once and then twice, correspondingto a hole of size 1 between 2 and 4 and one of size 2 between 5 and 8. The improvement withrespect to the notation of [8] lies in the fact that here we iterate over the elements of the set A rather than over all elements of [1; n ], such that we can avoid distinguishing several cases as in [8]. Definition 2.4.
The Askey–Wilson algebra of rank n −
2, denoted AW( n ), is the subalgebra of U q ( sl ) ⊗ n generated by all Λ A with A ⊆ [1; n ].We will refer to the algorithm described in Definition 2.3 as the right extension process, asto make the distinction with the following, alternative construction method. H. De Clercq An alternative method to associate to each A ⊆ [1; n ] an element of U q ( sl ) ⊗ n uses the leftcoideal comodule subalgebra I L and its coaction τ L , as introduced in Definition 2.2. Definition 2.5.
For any set A = { a , . . . , a m } ⊆ [1; n ], ordered such that a i < a i +1 for all i , wedefine (cid:98) Λ A ∈ U q ( sl ) ⊗ n by (cid:98) Λ A = 1 ⊗ ( a − ⊗ ←−− m − (cid:89) i =1 (cid:98) µ Ai (Λ) ⊗ ⊗ ( n − a m ) , with (cid:98) µ Ai = −−−−−−−−→ a m − a i − (cid:89) (cid:96) = a m − a i +1 +1 (cid:0) τ L ⊗ ⊗ (cid:96) (cid:1) (cid:0) ∆ ⊗ ⊗ ( a m − a i +1 ) (cid:1) , (2.10)where of course the term between brackets is absent if a i +1 = a i + 1. Example 2.2.
As before we take n = 9 and A = { , , , } , and find (cid:98) Λ { , , , } = 1 ⊗ (cid:0)(cid:98) µ { , , , } (cid:98) µ { , , , } (cid:98) µ { , , , } (cid:1) (Λ) ⊗ , with (cid:98) µ { , , , } = (cid:0) τ L ⊗ ⊗ (cid:1) ( τ L ⊗ (cid:98) µ { , , , } = ∆ ⊗ ⊗ and (cid:98) µ { , , , } = (cid:0) τ L ⊗ ⊗ (cid:1)(cid:0) ∆ ⊗ ⊗ (cid:1) .Again, the idea is that each element of A but the maximum a m corresponds to an applicationof ∆, whereas τ L creates the holes . However, as opposed to Definition 2.3, we now run throughthe elements of A in decreasing order, from right to left. This is why we refer to the algorithmof Definition 2.5 as the left extension process.Our first major task will be to prove the equivalence of the right and left extension processes,i.e., to show that they produce the same elements, for each set A . To do so, it will often beneeded to switch the order of certain algebra morphisms which act on mutually disjoint tensorproduct positions. More precisely, if X ∈ U q ( sl ) ⊗ and ϕ, ψ : U q ( sl ) → U q ( sl ) ⊗ , then we havethe following basic property:(1 ⊗ ⊗ ψ )( ϕ ⊗ X = ( ϕ ⊗ ⊗ ⊗ ψ ) X. (2.11) Remark 2.1.
This property also allows to replace the definitions (2.9) and (2.10) of µ Ai and (cid:98) µ Ai by certain equivalent expressions. For example, µ { , } = (cid:0) ⊗ ⊗ τ R (cid:1)(cid:0) ⊗ ⊗ τ R (cid:1) (1 ⊗ τ R )∆.Invoking (2.3) and (2.11), we have (cid:0) ⊗ ⊗ τ R (cid:1) (1 ⊗ τ R ) τ R = (cid:0) ⊗ ⊗ τ R (cid:1) (∆ ⊗ τ R = (cid:0) ∆ ⊗ ⊗ (cid:1) (1 ⊗ τ R ) τ R = (cid:0) ∆ ⊗ ⊗ (cid:1) (∆ ⊗ τ R , which allows to rewrite µ { , } , and of course this applies to any morphism of the form µ Ai with a i − a i − >
2. Similarly, one finds from (2.6) and (2.11) that (cid:0) τ L ⊗ ⊗ (cid:1) ( τ L ⊗ τ L = (cid:0) ⊗ ⊗ ∆ (cid:1) (1 ⊗ ∆) τ L and its generalizations for any (cid:98) µ Ai with a i +1 − a i > q -Bannai–Ito Algebra 7 Proposition 2.3.
The right and left extension processes produce exactly the same generators:for each A ⊆ [1; n ] one has Λ A = (cid:98) Λ A . Proof .
Without loss of generality, we can assume that min( A ) = 1 and max( A ) = n , sinceotherwise it suffices to add 1 in the remaining positions. We write A = [1; j ] ∪ [ i ; j ] ∪· · ·∪ [ i m ; n ]and proceed by induction on m . The case m = 1 follows from coassociativity:Λ [1; n ] = (cid:0) ⊗ ( n − ⊗ ∆ (cid:1) · · · (1 ⊗ ∆)∆(Λ) = (cid:0) ∆ ⊗ ⊗ ( n − (cid:1) · · · (∆ ⊗ (cid:98) Λ [1; n ] . Suppose now the claim is true for all sets consisting of m − (cid:98) A = [1; j ] ∪ [ i ; j ] ∪ · · · ∪ [ i m − ; j m − + 1]: Λ (cid:98) A = (cid:98) Λ (cid:98) A . The right extension process assertsΛ A = (cid:0) ⊗ ( n − ⊗ ∆ (cid:1) · · · (cid:0) ⊗ ( i m − ⊗ ∆ (cid:1)(cid:0) ⊗ ( i m − ⊗ τ R (cid:1) · · · (cid:0) ⊗ j m − ⊗ τ R (cid:1) Λ (cid:98) A . (2.12)On the other hand, by the left extension process we haveΛ (cid:98) A = (cid:98) Λ (cid:98) A = (cid:0) α j m − − ⊗ ⊗ ( j m − − (cid:1) · · · ( α ⊗ , (2.13)with each α i ∈ { ∆ , τ L } . When combining (2.12) and (2.13), it is clear that the second tensorproduct position in ∆(Λ) is left invariant by all the α i . Hence by (2.11) we may shift themorphisms in (2.12) through those in (2.13), such thatΛ A = (cid:0) α j m − − ⊗ ⊗ ( n − (cid:1) · · · (cid:0) α ⊗ ⊗ ( n − j m − ) (cid:1) Λ B , (2.14)with Λ B = (cid:0) ⊗ ( n − j m − − ⊗ ∆ (cid:1) · · · (cid:0) ⊗ ( i m − j m − ) ⊗ ∆ (cid:1)(cid:0) ⊗ ( i m − j m − − ⊗ τ R (cid:1) · · · (1 ⊗ τ R )∆(Λ) . By (2.7) and (2.11) we have(1 ⊗ ⊗ τ R )(1 ⊗ τ R )∆(Λ) = ( τ L ⊗ ⊗ ⊗ τ R )∆(Λ) = ( τ L ⊗ ⊗ τ L ⊗ . Repeating this, we findΛ B = (cid:0) ⊗ ( n − j m − − ⊗ ∆ (cid:1) · · · (cid:0) ⊗ ( i m − j m − ) ⊗ ∆ (cid:1)(cid:0) τ L ⊗ ⊗ ( i m − j m − − (cid:1) · · · ( τ L ⊗ . Invoking (2.11) again, we may shift all the 1 ⊗ (cid:96) ⊗ ∆ through all the τ L ⊗ ⊗ m , which, after usingcoassociativity, givesΛ B = (cid:0) τ L ⊗ ⊗ ( n − j m − − (cid:1) · · · (cid:0) τ L ⊗ ⊗ ( n − i m +1) (cid:1)(cid:0) ∆ ⊗ ⊗ ( n − i m ) (cid:1) · · · (∆ ⊗ (cid:98) Λ { }∪ [ i m − j m − +1; n − j m − +1] . Moreover, the left extension process and (2.13) imply that (cid:98) Λ A = (cid:0) α j m − − ⊗ ⊗ ( n − (cid:1) · · · (cid:0) α ⊗ ⊗ ( n − j m − ) (cid:1)(cid:98) Λ { }∪ [ i m − j m − +1; n − j m − +1] . The statement now follows from (2.14). (cid:4)
The reasoning established in the proof of Proposition 2.3 suggests the existence of severalother, so-called mixed extension processes, which produce the same elements Λ A . To be pre-cise, one can split the set A at any a j and perform the right extension process for the subset { a j +1 , . . . , a m } and consequently the left extension process for { a , . . . , a j − } . This is describedin the following definition. H. De Clercq Definition 2.6.
For any set A = { a , . . . , a m } ⊆ [1; n ], ordered such that a i < a i +1 for all i , wedefine Λ ( j ) A ∈ U q ( sl ) ⊗ n byΛ ( j ) A = 1 ⊗ ( a − ⊗ ←− j − (cid:89) i =1 (cid:98) µ Ai −−−−−−→ m (cid:89) i = j +1 µ Ai,j (Λ) ⊗ ⊗ ( n − a m ) , with (cid:98) µ Ai as in (2.10) and µ Ai,j = −−−−−−−→ a i − a j − (cid:89) (cid:96) = a i − − a j +1 (cid:0) ⊗ (cid:96) ⊗ τ R (cid:1) (cid:0) ⊗ ( a i − − a j ) ⊗ ∆ (cid:1) . By definition, we have Λ (1) A = Λ A and Λ ( m ) A = (cid:98) Λ A and following the proof of Proposition 2.3,we even have Λ ( j ) A = Λ A for all j ∈ { , . . . , m } . Example 2.3.
For n = 9, A = { , , , } and j = 2 we haveΛ (2) { , , , } = 1 ⊗ (cid:0)(cid:98) µ { , , , } µ { , , , } , µ { , , , } , (cid:1) (Λ) ⊗ , with µ { , , , } , = ∆, µ { , , , } , = (cid:0) ⊗ ⊗ τ R (cid:1)(cid:0) ⊗ ⊗ τ R (cid:1) (1 ⊗ ∆) and (cid:98) µ { , , , } = (cid:0) τ L ⊗ ⊗ (cid:1)(cid:0) ∆ ⊗ ⊗ (cid:1) . In the extension processes described above, the order in which we apply the different morphismsis of high importance. Nevertheless, there is some additional freedom in this order, which willcome in handy in many of the following proofs. More precisely, the upcoming Proposition 2.4asserts that when constructing Λ A , it suffices to first create all the holes between elements of A in order of appearance and then enlarge all holes and all intervals by applying repeatedly thecoproduct ∆. Proposition 2.4.
Let A = [1; j ] ∪ [ i ; j ] ∪ · · · ∪ [ i k ; j k ] , then one has Λ A = ←−− k − (cid:89) n =0 −−−−→ β n, i , j (cid:89) (cid:96) = α n, i , j (1 ⊗ n ⊗ ∆ ⊗ ⊗ (cid:96) ) Λ { , , ,..., k − } , (2.15) where α n, i , j = (cid:40) j k − j m if n = 2 m − is even ,j k − i m + 1 if n = 2 m − is oddand β n, i , j = (cid:40) j k − i m − if n = 2 m − is even ,j k − j m − − if n = 2 m − is oddand where we set i = (1 , i , . . . , i k ) and j = ( j , j , . . . , j k ) . igher Rank Relations for the Askey–Wilson and q -Bannai–Ito Algebra 9 Proof .
By induction on k , the case k = 1 being trivial by coassociativity. Suppose hence theclaim holds for all sets consisting of k − (cid:98) A = [1; j ] ∪ [ i ; j ] ∪ · · · ∪ [ i k − ; j k − ]:Λ (cid:98) A = ←−− k − (cid:89) n =0 −−−−→ β n, (cid:101) i , (cid:101) j (cid:89) (cid:96) = α n, (cid:101) i , (cid:101) j (cid:0) ⊗ n ⊗ ∆ ⊗ ⊗ (cid:96) (cid:1) Λ { , , ,..., k − } , (2.16)where (cid:101) i = (1 , i , . . . , i k − ) and (cid:101) j = ( j , j , . . . , j k − ). From the right extension process, usingcoassociativity and Remark 2.1, it is clear thatΛ A = (cid:0) ⊗ ( i k − ⊗ ∆ ⊗ ⊗ ( j k − i k − (cid:1) · · · (cid:0) ⊗ ( i k − ⊗ ∆ (cid:1)(cid:0) ⊗ j k − ⊗ ∆ ⊗ ⊗ ( i k − j k − − (cid:1) · · ·× (cid:0) ⊗ j k − ⊗ ∆ ⊗ (cid:1)(cid:0) ⊗ j k − ⊗ τ R (cid:1)(cid:0) ⊗ ( j k − − ⊗ ∆ (cid:1) Λ (cid:98) A . (2.17)Observe that when combining (2.16) and (2.17), the morphisms in (2.16) with n ≤ k − { , , ,..., k − } of lower index than the morphisms in the secondline in (2.17). We may hence switch their order as in (2.11): (cid:0) ⊗ j k − ⊗ τ R (cid:1)(cid:0) ⊗ ( j k − − ⊗ ∆ (cid:1) Λ (cid:98) A = ←−− k − (cid:89) n =0 −−−−−→ β n, (cid:101) i , (cid:101) j +2 (cid:89) (cid:96) = α n, (cid:101) i , (cid:101) j +2 (cid:0) ⊗ n ⊗ ∆ ⊗ ⊗ (cid:96) (cid:1) Λ B , (2.18)where Λ B is defined as χ (Λ { , , ,..., k − } ), where χ is the morphism (cid:0) ⊗ ( j k − − i k − +2 k − ⊗ τ R (cid:1)(cid:0) ⊗ ( j k − − i k − +2 k − ⊗ ∆ (cid:1) −−−−−−−→ j k − − i k − − (cid:89) (cid:96) =0 (cid:0) ⊗ (2 k − ⊗ ∆ ⊗ ⊗ (cid:96) (cid:1) . Here, the term between brackets corresponds to n = 2 k − α k − , (cid:101) i , (cid:101) j = 0 and β k − , (cid:101) i , (cid:101) j = j k − − i k − − . (2.19)Using coassociativity, separating the term for (cid:96) = 0 and relying on the right extension process,we haveΛ B = (cid:0) ⊗ ( j k − − i k − +2 k − ⊗ τ R (cid:1) −−−−−→ j k − − i k − (cid:89) (cid:96) =1 (cid:0) ⊗ (2 k − ⊗ ∆ ⊗ ⊗ (cid:96) (cid:1) Λ { , , ,..., k − , k − } . Now it is manifest that all ∆ in the product between brackets act on the tensor product position2 k − { , , ,..., k − , k − } , whereas τ R in fact acts on the last position 2 k −
2. Hence we mayagain apply (2.11) to switch the order:Λ B = −−−−−−−→ j k − − i k − +1 (cid:89) (cid:96) =2 (cid:0) ⊗ (2 k − ⊗ ∆ ⊗ ⊗ (cid:96) (cid:1) Λ { , , ,..., k − , k − } . Note that the lower and upper bounds in the product between brackets equal α k − , (cid:101) i , (cid:101) j + 2 and β k − , (cid:101) i , (cid:101) j + 2 respectively, by (2.19).Combined with (2.17) and (2.18), this yieldsΛ A = (cid:0) ⊗ ( i k − ⊗ ∆ ⊗ ⊗ ( j k − i k − (cid:1) · · · (cid:0) ⊗ ( i k − ⊗ ∆ (cid:1)(cid:0) ⊗ j k − ⊗ ∆ ⊗ ⊗ ( i k − j k − − (cid:1) · · · × (cid:0) ⊗ j k − ⊗ ∆ ⊗ (cid:1) ←−− k − (cid:89) n =0 −−−−−→ β n, (cid:101) i , (cid:101) j +2 (cid:89) (cid:96) = α n, (cid:101) i , (cid:101) j +2 (cid:0) ⊗ n ⊗ ∆ ⊗ ⊗ (cid:96) (cid:1) Λ { , , ,..., k − , k − } . Moreover, all morphisms in the first line act on the tensor product positions 2 k − k − { , , ,..., k − , k − } , whereas those on the second line act on positions 1 to 2 k −
3. The ordercan hence be switched again by (2.11), leading first toΛ A = ←−− k − (cid:89) n =0 −−−−−−−−−−→ β n, (cid:101) i , (cid:101) j + j k − j k − (cid:89) (cid:96) = α n, (cid:101) i , (cid:101) j + j k − j k − (cid:0) ⊗ n ⊗ ∆ ⊗ ⊗ (cid:96) (cid:1) × (cid:0) ⊗ ( i k − j k − +2 k − ⊗ ∆ ⊗ ⊗ ( j k − i k − (cid:1) · · · (cid:0) ⊗ ( i k − j k − +2 k − ⊗ ∆ (cid:1) × (cid:0) ⊗ (2 k − ⊗ ∆ ⊗ ⊗ ( i k − j k − − (cid:1) · · · (cid:0) ⊗ (2 k − ⊗ ∆ ⊗ (cid:1) Λ { , , ,..., k − , k − } and then, after switching the morphisms on the second with those on the third line, toΛ A = ←−− k − (cid:89) n =0 −−−−−−−−−−→ β n, (cid:101) i , (cid:101) j + j k − j k − (cid:89) (cid:96) = α n, (cid:101) i , (cid:101) j + j k − j k − (cid:0) ⊗ n ⊗ ∆ ⊗ ⊗ (cid:96) (cid:1) × (cid:0) ⊗ (2 k − ⊗ ∆ ⊗ ⊗ ( j k − j k − − (cid:1) · · · (cid:0) ⊗ (2 k − ⊗ ∆ ⊗ ⊗ ( j k − i k +1) (cid:1) × (cid:0) ⊗ (2 k − ⊗ ∆ ⊗ ⊗ ( j k − i k − (cid:1) · · · (cid:0) ⊗ (2 k − ⊗ ∆ (cid:1) Λ { , , ,..., k − , k − } . Now observe that for every n ∈ { , , . . . , k − } one has α n, i , j = α n, (cid:101) i , (cid:101) j + j k − j k − and β n, i , j = β n, (cid:101) i , (cid:101) j + j k − j k − , and moreover we have α k − , i , j = j k − i k +1, β k − , i , j = j k − j k − − α k − , i , j = 0and β k − , i , j = j k − i k −
1. Hence the expression above coincides with (2.15). (cid:4)
In this section, we formulate the main results of this paper: the algebraic relations satisfied inthe higher rank Askey–Wilson algebra AW( n ). As in the rank one case, these will be of the form[Λ A , Λ B ] q = (cid:0) q − − q (cid:1) Λ ( A ∪ B ) \ ( A ∩ B ) + (cid:0) q − q − (cid:1)(cid:0) Λ A ∩ B Λ A ∪ B + Λ A \ ( A ∩ B ) Λ B \ ( A ∩ B ) (cid:1) , ( ∗ )or [Λ A , Λ B ] = 0 , (∆)under suitable conditions on the sets A and B . In this section, we present these conditions,which, based on extensive computer calculations, we believe to be minimal. They can be statedas follows. Theorem 3.1.
Let
A, B ⊆ [1; n ] be such that B ⊆ A , then Λ A and Λ B commute. Section 6 will be devoted to the proof of this theorem.
Definition 3.1.
For
A, B ⊆ [1; n ] we write A ≺ B if max( A ) < min( B ) or if either A or B isempty. Theorem 3.2.
Let A , A , A and A be ( potentially empty ) subsets of [1; n ] satisfying A ≺ A ≺ A ≺ A . igher Rank Relations for the Askey–Wilson and q -Bannai–Ito Algebra 11 The standard relation [Λ A , Λ B ] q = (cid:0) q − − q (cid:1) Λ ( A ∪ B ) \ ( A ∩ B ) + (cid:0) q − q − (cid:1)(cid:0) Λ A ∩ B Λ A ∪ B + Λ A \ ( A ∩ B ) Λ B \ ( A ∩ B ) (cid:1) is satisfied for A and B defined by one of the following relations: A = A ∪ A ∪ A , B = A ∪ A , (3.1) A = A ∪ A , B = A ∪ A ∪ A , (3.2) A = A ∪ A ∪ A , B = A ∪ A ∪ A . (3.3)This will be shown in Section 7.Our general strategy to prove a relation of the form ( ∗ ) will be as follows. First we willconstruct an operator χ , by combining morphisms of the form 1 ⊗ n ⊗ α ⊗ ⊗ m , α ∈ { ∆ , τ R , τ L } and n, m ∈ N , such that χ (Λ A (cid:48) ) = Λ A , χ (Λ B (cid:48) ) = Λ B , χ (Λ ( A (cid:48) ∪ B (cid:48) ) \ ( A (cid:48) ∩ B (cid:48) ) ) = Λ ( A ∪ B ) \ ( A ∩ B ) , . . . for certain (less complicated) sets A (cid:48) , B (cid:48) . To prove ( ∗ ) it now suffices to show[Λ A (cid:48) , Λ B (cid:48) ] q = (cid:0) q − − q (cid:1) Λ ( A (cid:48) ∪ B (cid:48) ) \ ( A (cid:48) ∩ B (cid:48) ) + (cid:0) q − q − (cid:1)(cid:0) Λ A (cid:48) ∩ B (cid:48) Λ A (cid:48) ∪ B (cid:48) + Λ A (cid:48) \ ( A (cid:48) ∩ B (cid:48) ) Λ B (cid:48) \ ( A (cid:48) ∩ B (cid:48) ) (cid:1) , ( ∗∗ )apply the operator χ to both sides of the equation and use its linearity and multiplicativity.In this case we will write “( ∗ ) follows from ( ∗∗ ) by χ ”. The same strategy applies to relationsof the form (∆). In this respect, the relations of Theorem 3.2 can in fact be derived from thefollowing 9 fundamental cases. Proposition 3.1.
The standard relation ( ∗ ) holds for the following combinations of sets A and B : A = { , , , , . . . , k } ,B = { , , , . . . , k, k + 1 } ; (C1) A = { , , , . . . , k, k + 1 } ,B = { , k + 1 } ; (C2) A = { , k + 1 } ,B = { , , , , . . . , k } ; (C3) A = { , , , , . . . , k, k + 2 (cid:96) + 2 } ,B = { , , , . . . , k, k + 1 , k + 3 , . . . , k + 2 (cid:96) + 1 } ; (C4) A = { , , , , . . . , k, k + 2 (cid:96) + 3 } ,B = { , , , . . . , k, k + 2 , k + 4 , . . . , k + 2 (cid:96) + 2 } ; (C4 (cid:48) ) A = { , , , . . . , k, k + 1 , k + 3 , . . . , k + 2 (cid:96) + 1 } ,B = { , k + 1 , k + 3 , . . . , k + 2 (cid:96) + 1 , k + 2 (cid:96) + 2 } ; (C5) A = { , , , . . . , k, k + 2 , k + 4 , . . . , k + 2 (cid:96) + 2 } ,B = { , k + 2 , k + 4 , . . . , k + 2 (cid:96) + 2 , k + 2 (cid:96) + 3 } ; (C5 (cid:48) ) A = { , k + 1 , k + 3 , . . . , k + 2 (cid:96) + 1 , k + 2 (cid:96) + 2 } ,B = { , , , , . . . , k, k + 2 (cid:96) + 2 } ; (C6) A = { , k + 2 , k + 4 , . . . , k + 2 (cid:96) + 2 , k + 2 (cid:96) + 3 } ,B = { , , , , . . . , k, k + 2 (cid:96) + 3 } , (C6 (cid:48) ) with k, (cid:96) ∈ N . Throughout the whole paper, it will turn out useful to switch orders in nested q -commutators.A straightforward calculation gives the following: Lemma 4.1.
Let A be any algebra and α , β , γ , δ elements of A , then one has [ α, [ γ, δ ] q ] q = [[ α, γ ] q , δ ] q if [ α, δ ] = 0 , (4.1)[ α, [ γ, δ ] q ] q = [ γ, [ α, δ ] q ] q if [ α, γ ] = 0 , (4.2)[[ γ, δ ] q , β ] q = [[ γ, β ] q , δ ] q if [ β, δ ] = 0 , (4.3)[[ γ, δ ] q , β ] q = [ γ, [ δ, β ] q ] q if [ β, γ ] = 0 . (4.4)The following commutation relations are so natural that they will often be relied on in proofsin Subsection 4.2 without explicit reference. Lemma 4.2.
For any i ∈ [1; n ] and any subset A ⊆ [1; n ] one has [Λ A , Λ { i } ] = 0 . Proof .
This is immediate by the fact that Λ { i } = 1 ⊗ ( i − ⊗ Λ ⊗ ⊗ ( n − i ) and that Λ is the Casimiroperator of U q ( sl ). (cid:4) Lemma 4.3.
Let
A, A (cid:48) ⊆ [1; n ] be such that A ≺ A (cid:48) , where we use Definition , then one has [Λ A , Λ A (cid:48) ] = 0 , [Λ A , Λ A ∪ A (cid:48) ] = 0 , [Λ A (cid:48) , Λ A ∪ A (cid:48) ] = 0 . Proof .
The first statement is trivial, since Λ A and Λ A (cid:48) live in disjoint tensor product positions.For the second claim, writing A = [1; j ] ∪ [ i ; j ] ∪ · · · ∪ [ i k ; j k ] , A (cid:48) = [ i k +1 ; j k +1 ] ∪ · · · ∪ [ i k + (cid:96) ; j k + (cid:96) ] , the statement follows from [Λ { } , Λ { , } ] = 0 by a suitable morphism of the form χ = −−−−−−→ j k + (cid:96) − (cid:89) m = i k +1 − (cid:0) ⊗ m ⊗ β m (cid:1) −−−→ i k +1 − (cid:89) m =1 (cid:0) α m ⊗ ⊗ m (cid:1) , where each α m ∈ { ∆ , τ L } and each β m ∈ { ∆ , τ R } . The third statement follows analogously. (cid:4) The next lemma provides a first generalization of the relations (1.2)–(1.4).
Lemma 4.4.
Let i ∈ [1; n ] and A , A ⊆ [1; n ] be such that A ≺ { i } ≺ A , then the relation ( ∗ ) holds for ( A, B ) one of the couples ( A ∪ { i } , { i } ∪ A ) , ( { i } ∪ A , A ∪ A ) , ( A ∪ A , A ∪ { i } ) . igher Rank Relations for the Askey–Wilson and q -Bannai–Ito Algebra 13 Proof .
Let A ∪ { i } ∪ A = { a , . . . , a m } , ordered such that a (cid:96) < a (cid:96) +1 for all (cid:96) and let j be suchthat a j = i . The mixed extension process with parameter j asserts the existence of a morphism χ which sends { } (cid:55)→ A , { } (cid:55)→ { i } , { } (cid:55)→ A . Hence the statements follow from (1.2)–(1.4) by χ . (cid:4) A final immediate commutation relation is the following.
Lemma 4.5.
For any k ∈ N one has [Λ { , ,..., k } , Λ { , k +1 } ] = 0 . Proof .
By induction on k . The case k = 1 is trivial by Lemma 4.2. Suppose hence the claimhas been proven for k −
1. Lemma 4.4 assertsΛ { , , ,..., k } = [Λ { , } , Λ { , , ,..., k } ] q q − − q + Λ { } Λ { , , , ,..., k } + Λ { } Λ { , ,..., k } q + q − , hence it suffices to show that Λ { , k +1 } commutes with each term in the right-hand side.For Λ { , } this follows from [Λ { , } , Λ { } ] = 0 by χ = (cid:0) ⊗ (2 k − ⊗ τ R ) · · · (cid:0) ⊗ ⊗ τ R (cid:1) (1 ⊗ ∆ ⊗ χ = (cid:0) τ L ⊗ ⊗ (2 k − (cid:1)(cid:0) ⊗ ∆ ⊗ ⊗ (2 k − (cid:1) , χ = (cid:0) ⊗ ∆ ⊗ ⊗ (2 k − (cid:1)(cid:0) ⊗ ∆ ⊗ ⊗ (2 k − (cid:1) and χ = (cid:0) τ L ⊗ ⊗ (2 k − (cid:1)(cid:0) τ L ⊗ ⊗ (2 k − (cid:1) respectively. (cid:4) (cid:48) ) In this section we will prove that the relation ( ∗ ) is satisfied for the combinations of sets (C1)–(C6 (cid:48) ). We will work out the proof for three of these cases in detail, and describe concisely howone can show the remaining cases. Lemma 4.6.
The relation ( ∗ ) holds for the sets (C2) and (C5) with (cid:96) = 0 . Proof .
We will prove both claims together in one single induction on k . For k = 1 the firstclaim coincides with (1.3) and the second follows by direct calculation.Our induction hypothesis states that ( ∗ ) holds for A = { , , . . . , k − , k − } , B = { , k − } (4.5)and A = { , , . . . , k − , k − } , B = { , k − , k } . (4.6)By χ = (cid:0) ⊗ (2 k − ⊗ ∆ ⊗ ⊗ (cid:1)(cid:0) ⊗ (2 k − ⊗ ∆ ⊗ (cid:1) , (4.5) implies ( ∗ ) for A = { , , . . . , k − , k − , k, k + 1 } , B = { , k + 1 } (4.7)and by χ = 1 ⊗ (2 k − ⊗ ∆, (4.6) gives rise to ( ∗ ) for A = { , , . . . , k − , k − } , B = { , k − , k, k + 1 } . (4.8)We will first compute[Λ { , , ,..., k, k +1 } , Λ { , k +1 } ] q . (4.9)4 H. De ClercqBy Lemma 4.5 we have (∆) for A = { , k − } , B = { , , . . . , k − } , which, upon applying χ = (cid:0) ⊗ (2 k − ⊗ τ R (cid:1)(cid:0) ⊗ (2 k − ⊗ ∆ ⊗ (cid:1) , yields (∆) for A = { , k + 1 } , B = { , , . . . , k − , k − } , (4.10)whereas by χ = (cid:0) ⊗ (2 k − ⊗ τ R (cid:1)(cid:0) ⊗ (2 k − ⊗ τ R (cid:1) we have (∆) for A = { , k + 1 } , B = { , , . . . , k − } . (4.11)By Lemma 4.4 we may writeΛ { , ,..., k, k +1 } = [Λ { , ,..., k − , k − } , Λ { k − , k, k +1 } ] q q − − q + Λ { k − } Λ { , ,..., k − , k − , k, k +1 } + Λ { , ,..., k − } Λ { k, k +1 } q + q − . Substituting this in (4.9) and using (4.4) by (4.10), (4.9) becomes[Λ { , ,..., k − , k − } , [Λ { k − , k, k +1 } , Λ { , k +1 } ] q ] q q − − q + Λ { k − } [Λ { , ,..., k − , k − , k, k +1 } , Λ { , k +1 } ] q + Λ { , ,..., k − } [Λ { k, k +1 } , Λ { , k +1 } ] q q + q − , where we have used (4.11) and Lemma 4.3.The relation ( ∗ ) holds for A = { k − , k, k + 1 } , B = { , k + 1 } and for A = { k, k + 1 } , B = { , k + 1 } , as one sees by applying χ = (cid:0) τ L ⊗ ⊗ (2 k − (cid:1) · · · (cid:0) τ L ⊗ ⊗ (cid:1) (1 ⊗ ∆ ⊗
1) resp. χ = (cid:0) τ L ⊗ ⊗ (2 k − (cid:1) · · · (cid:0) τ L ⊗ ⊗ (cid:1) to (1.3). With this and (4.7), (4.9) becomes[Λ { , ,..., k − , k − } , Λ { , k − , k } ] q − Λ { k +1 } [Λ { , ,..., k − , k − } , Λ { , k − , k, k +1 } ] q + Λ { } [Λ { , ,..., k − , k − } , Λ { k − , k } ] q q + q − − (cid:0) q − q − (cid:1) Λ { k − } Λ { , , ,..., k − , k − , k } + q − q − q + q − Λ { k − } (cid:0) Λ { k +1 } Λ { , , ,..., k − , k − , k, k +1 } + Λ { } Λ { , ,..., k − , k − , k } (cid:1) − (cid:0) q − q − (cid:1) Λ { , ,..., k − } Λ { , k } + q − q − q + q − Λ { , ,..., k − } (cid:0) Λ { k +1 } Λ { , k, k +1 } + Λ { } Λ { k } (cid:1) . The first q -commutator can be expanded by (4.6), the second by (4.8), the third by Lemma 4.4.Writing everything down, a lot of common terms will cancel, eventually leading to[Λ { , , ,..., k, k +1 } , Λ { , k +1 } ] q = (cid:0) q − − q (cid:1) Λ { , , ,..., k } + (cid:0) q − q − (cid:1)(cid:0) Λ { k +1 } Λ { , , ,..., k, k +1 } + Λ { } Λ { , ,..., k } (cid:1) . (4.12)This proves the first part of the claim.By χ = 1 ⊗ (2 k − ⊗ ∆ ⊗ χ = 1 ⊗ k ⊗ τ R , (4.12) implies that ( ∗ ) holds for A = { , , . . . , k, k + 1 , k + 2 } , B = { , k + 2 } (4.13)and A = { , , . . . , k, k + 2 } , B = { , k + 2 } . (4.14)igher Rank Relations for the Askey–Wilson and q -Bannai–Ito Algebra 15Applying χ = (cid:0) τ L ⊗ ⊗ k (cid:1) · · · (cid:0) τ L ⊗ ⊗ (cid:1) to (1.3), we may writeΛ { , k +1 } = [Λ { k +1 , k +2 } , Λ { , k +2 } ] q q − − q + Λ { k +2 } Λ { , k +1 , k +2 } + Λ { k +1 } Λ { } q + q − . (4.15)We have (∆) for A = { , k + 2 } and B = { , , . . . , k, k + 1 } , as follows from Lemma 4.5 by χ = 1 ⊗ (2 k − ⊗ ∆ ⊗
1. Substituting (4.15) in (4.9) and using (4.1), (4.9) becomes[[Λ { , ,..., k, k +1 } , Λ { k +1 , k +2 } ] q , Λ { , k +2 } ] q q − − q + Λ { k +2 } [Λ { , ,..., k, k +1 } , Λ { , k +1 , k +2 } ] q + (cid:0) q − q − (cid:1) Λ { } Λ { k +1 } Λ { , ,..., k, k +1 } q + q − . The relation ( ∗ ) holds for A = { , , . . . , k, k + 1 } and B = { k + 1 , k + 2 } by Lemma 4.4.Hence (4.9) becomes[Λ { , ,..., k, k +2 } , Λ { , k +2 } ] q − Λ { k +1 } [Λ { , ,..., k, k +1 , k +2 } , Λ { , k +2 } ] q + (cid:0) q − q − (cid:1) Λ { k +2 } Λ { , ,..., k } Λ { , k +2 } q + q − + Λ { k +2 } [Λ { , ,..., k, k +1 } , Λ { , k +1 , k +2 } ] q + (cid:0) q − q − (cid:1) Λ { } Λ { k +1 } Λ { , ,..., k, k +1 } q + q − , (4.16)where we have used (∆) for A = { , , . . . , k } , B = { , k + 2 } , which follows from Lemma 4.5by χ = 1 ⊗ k ⊗ τ R . The first q -commutator can be expanded by (4.14), the second by (4.13). Onthe other hand, we already know an expression for (4.9), namely (4.12). Comparing these, theonly remaining q -commutator in (4.16) can be expanded as[Λ { , ,..., k, k +1 } , Λ { , k +1 , k +2 } ] q = (cid:0) q − − q (cid:1) Λ { , , ,..., k, k +2 } + (cid:0) q − q − (cid:1)(cid:0) Λ { k +1 } Λ { , , ,..., k, k +1 , k +2 } + Λ { , k +2 } Λ { , ,..., k } (cid:1) . This concludes the induction. (cid:4)
By a completely analogous inductive proof, one can show the following.
Lemma 4.7.
The relation ( ∗ ) holds for the sets (C3) and (C4) with k = 1 . Somewhat different is our strategy to prove the relation ( ∗ ) for the sets (C4) and (C4 (cid:48) ). Lemma 4.8.
The standard relation ( ∗ ) holds for the sets (C4) and (C4 (cid:48) ) , i.e., for A = { , , , , . . . , k, k + 2 (cid:96) + 2 + δ } ,B = { , , , . . . , k, k + 1 + δ, k + 3 + δ, . . . , k + 2 (cid:96) + 1 + δ } , with k, (cid:96) ∈ N and δ ∈ { , } . Proof .
We need to rewrite[Λ { , , , ,..., k, k +2 (cid:96) +2+ δ } , Λ { , , ,..., k, k +1+ δ, k +3+ δ,..., k +2 (cid:96) +1+ δ } ] q . (4.17)First observe that[Λ { , k +2 (cid:96) +3+ δ } , Λ { , , ,..., k, k +1+ δ, k +3+ δ,..., k +2 (cid:96) +1+ δ } ] = 0 . (4.18)6 H. De ClercqThis follows from Lemma 4.5 with k + (cid:96) + δ instead of k , by χ = (cid:0) ⊗ (2 k − ⊗ ∆ ⊗ ⊗ (2 (cid:96) +2) (cid:1) − δ (cid:0) ⊗ (2 k +2 (cid:96) +2 δ ) ⊗ τ R (cid:1) . The relation ( ∗ ) for (C2) with k + 1 instead of k , acted upon with χ = (cid:0) ⊗ k ⊗ ∆ ⊗ ⊗ (2 (cid:96) +1+ δ ) (cid:1) · · · (cid:0) ⊗ k ⊗ ∆ ⊗ ⊗ (cid:1) , gives rise to the identityΛ { , , , ,..., k, k +2 (cid:96) +2+ δ } = [Λ { , , ,..., k, k +2 (cid:96) +2+ δ, k +2 (cid:96) +3+ δ } , Λ { , k +2 (cid:96) +3+ δ } ] q q − − q + Λ { k +2 (cid:96) +3+ δ } Λ { , , ,..., k, k +2 (cid:96) +2+ δ, k +2 (cid:96) +3+ δ } + Λ { } Λ { , ,..., k, k +2 (cid:96) +2+ δ } q + q − . Using (4.3) by (4.18), we may write (4.17) as[[Λ { , ,..., k, k +2 (cid:96) +2+ δ, k +2 (cid:96) +3+ δ } , Λ { , ,..., k, k +1+ δ, k +3+ δ,..., k +2 (cid:96) +1+ δ } ] q , Λ { , k +2 (cid:96) +3+ δ } ] q q − − q + Λ { k +2 (cid:96) +3+ δ } [Λ { , , ,..., k, k +2 (cid:96) +2+ δ, k +2 (cid:96) +3+ δ } , Λ { , ,..., k, k +1+ δ, k +3+ δ,..., k +2 (cid:96) +1+ δ } ] q q + q − + Λ { } [Λ { , ,..., k, k +2 (cid:96) +2+ δ } , Λ { , ,..., k, k +1+ δ, k +3+ δ,..., k +2 (cid:96) +1+ δ } ] q q + q − . (4.19)The relation ( ∗ ) is satisfied for A = { , , , . . . , k, k + 2 (cid:96) + 2 + δ } ,B = { , , , . . . , k, k + 1 + δ, k + 3 + δ, . . . , k + 2 (cid:96) + 1 + δ } , as follows from (C3) with (cid:96) + 1 instead of k , by χ = −−−−−−→ k +2 (cid:96) − δ (cid:89) n =2 (cid:96) +2+ δ (cid:0) τ L ⊗ ⊗ ( n +1) (cid:1)(cid:0) ∆ ⊗ ⊗ n (cid:1) (cid:0) τ L ⊗ ⊗ (2 (cid:96) +2) (cid:1) δ and putting a factor 1 ⊗ in front. By χ = 1 ⊗ (2 k +2 (cid:96) +1+ δ ) ⊗ ∆, we also have ( ∗ ) for A = { , , , . . . , k, k + 2 (cid:96) + 2 + δ, k + 2 (cid:96) + 3 + δ } ,B = { , , , . . . , k, k + 1 + δ, k + 3 + δ, . . . , k + 2 (cid:96) + 1 + δ } . This helps us expand the first and third line of (4.19), such that (4.17) becomes[Λ { k +1+ δ, k +3+ δ,..., k +2 (cid:96) +1+ δ, k +2 (cid:96) +2+ δ, k +2 (cid:96) +3+ δ } , Λ { , k +2 (cid:96) +3+ δ } ] q − Λ { , ..., k } q + q − [Λ { , ,..., k, k +1+ δ, k +3+ δ,..., k +2 (cid:96) +1+ δ, k +2 (cid:96) +2+ δ, k +2 (cid:96) +3+ δ } , Λ { , k +2 (cid:96) +3+ δ } ] q − Λ { k +1+ δ, k +3+ δ,..., k +2 (cid:96) +1+ δ } q + q − [Λ { k +2 (cid:96) +2+ δ, k +2 (cid:96) +3+ δ } , Λ { , k +2 (cid:96) +3+ δ } ] q + Λ { k +2 (cid:96) +3+ δ } q + q − [Λ { , , ,..., k, k +2 (cid:96) +2+ δ, k +2 (cid:96) +3+ δ } , Λ { , ,..., k, k +1+ δ, k +3+ δ,..., k +2 (cid:96) +1+ δ } ] q − (cid:0) q − q − (cid:1) Λ { } Λ { k +1+ δ, k +3+ δ,..., k +2 (cid:96) +1+ δ, k +2 (cid:96) +2+ δ } + q − q − q + q − Λ { } Λ { , ,..., k } Λ { , ,..., k, k +1+ δ, k +3+ δ,..., k +2 (cid:96) +1+ δ, k +2 (cid:96) +2+ δ } igher Rank Relations for the Askey–Wilson and q -Bannai–Ito Algebra 17+ q − q − q + q − Λ { } Λ { k +2 (cid:96) +2+ δ } Λ { k +1+ δ, k +3+ δ,..., k +2 (cid:96) +1+ δ } . (4.20)Here we have used the commutation relations[Λ { , ,..., k } , Λ { , k +2 (cid:96) +3+ δ } ] = 0 , [Λ { , ,..., k } , Λ { , ,..., k, k +1+ δ, k +3+ δ,..., k +2 (cid:96) +1+ δ, k +2 (cid:96) +2+ δ, k +2 (cid:96) +3+ δ } ] = 0 , [Λ { k +1+ δ, k +3+ δ,..., k +2 (cid:96) +1+ δ } , Λ { k +2 (cid:96) +2+ δ, k +2 (cid:96) +3+ δ } ] = 0 , [Λ { k +1+ δ, k +3+ δ,..., k +2 (cid:96) +1+ δ } , Λ { , k +2 (cid:96) +3+ δ } ] = 0 , which follow from Lemmas 4.3 and 4.5.The relation ( ∗ ) holds for the sets A = { k + 1 + δ, k + 3 + δ, . . . , k + 2 (cid:96) + 1 + δ, k + 2 (cid:96) + 2 + δ, k + 2 (cid:96) + 3 + δ } ,B = { , k + 2 (cid:96) + 3 + δ } ; (4.21) A = { k + 2 (cid:96) + 2 + δ, k + 2 (cid:96) + 3 + δ } ,B = { , k + 2 (cid:96) + 3 + δ } ; (4.22) A = { , , . . . , k, k + 1 + δ, k + 3 + δ, . . . , k + 2 (cid:96) + 1 + δ, k + 2 (cid:96) + 2 + δ, k + 2 (cid:96) + 3 + δ } ,B = { , k + 2 (cid:96) + 3 + δ } . (4.23)For (4.21), this follows from (C2) with (cid:96) + 1 instead of k , upon applying χ = −−−−−−→ k +2 (cid:96) +1+ δ (cid:89) n =2 (cid:96) +3 (cid:0) τ L ⊗ ⊗ n (cid:1) (cid:0) ⊗ (2 (cid:96) +1) ⊗ ∆ ⊗ (cid:1) . The claim for (4.22) follows from (1.3), by χ = −−−−−−→ k +2 (cid:96) +1+ δ (cid:81) n =2 (cid:0) τ L ⊗ ⊗ n (cid:1) , and finally (4.23) followsfrom (C2) with k + (cid:96) + δ instead of k , by χ = (cid:0) ⊗ (2 k − ⊗ ∆ ⊗ ⊗ (2 (cid:96) +2) (cid:1) − δ (cid:0) ⊗ (2 k +2 (cid:96) − δ ) ⊗ ∆ ⊗ (cid:1) .Bringing the fourth line of (4.20) to the left-hand side and expanding everything as explained,we find[Λ { , , , ,..., k, k +2 (cid:96) +2+ δ } , Λ { , , ,..., k, k +1+ δ, k +3+ δ,..., k +2 (cid:96) +1+ δ } ] q − Λ { k +2 (cid:96) +3+ δ } q + q − [Λ { , , ,..., k, k +2 (cid:96) +2+ δ, k +2 (cid:96) +3+ δ } , Λ { , ,..., k, k +1+ δ, k +3+ δ,..., k +2 (cid:96) +1+ δ } ] q = (cid:0) q − − q (cid:1) Λ { , k +1+ δ, k +3+ δ,..., k +2 (cid:96) +1+ δ, k +2 (cid:96) +2+ δ } + (cid:0) q − q − (cid:1) Λ { , ,..., k } Λ { , , ,..., k, k +1+ δ, k +3+ δ,..., k +2 (cid:96) +1+ δ, k +2 (cid:96) +2+ δ } + (cid:0) q − q − (cid:1) Λ { , k +2 (cid:96) +2+ δ } Λ { k +1+ δ, k +3+ δ,..., k +2 (cid:96) +1+ δ } + (cid:0) q − q − (cid:1) Λ { k +2 (cid:96) +3+ δ } Λ { , k +1+ δ, k +3+ δ,..., k +2 (cid:96) +1+ δ, k +2 (cid:96) +2+ δ, k +2 (cid:96) +3+ δ } − q − q − q + q − Λ { k +2 (cid:96) +3+ δ } Λ { , ,..., k } Λ { , , ,..., k, k +1+ δ, k +3+ δ,..., k +2 (cid:96) +1+ δ, k +2 (cid:96) +2+ δ, k +2 (cid:96) +3+ δ } − q − q − q + q − Λ { k +2 (cid:96) +3+ δ } Λ { , k +2 (cid:96) +2+ δ, k +2 (cid:96) +3+ δ } Λ { k +1+ δ, k +3+ δ,..., k +2 (cid:96) +1+ δ } . Now let us denote by Θ and Ξ respectively the expressionsΘ = [Λ { , , , ,..., k, k +2 (cid:96) +2+ δ } , Λ { , , ,..., k, k +1+ δ, k +3+ δ,..., k +2 (cid:96) +1+ δ } ] q , (cid:0) q − − q (cid:1) Λ { , k +1+ δ, k +3+ δ,..., k +2 (cid:96) +1+ δ, k +2 (cid:96) +2+ δ } + (cid:0) q − q − (cid:1) Λ { , ,..., k } Λ { , , ,..., k, k +1+ δ, k +3+ δ,..., k +2 (cid:96) +1+ δ, k +2 (cid:96) +2+ δ } + (cid:0) q − q − (cid:1) Λ { , k +2 (cid:96) +2+ δ } Λ { k +1+ δ, k +3+ δ,..., k +2 (cid:96) +1+ δ } , both in the (2 k + 2 (cid:96) + 2 + δ )-fold tensor product. Then the relation above becomesΘ ⊗ − q + q − Λ { k +2 (cid:96) +3+ δ } (cid:0) ⊗ (2 k +2 (cid:96) +1+ δ ) ⊗ ∆ (cid:1) Θ= Ξ ⊗ − q + q − Λ { k +2 (cid:96) +3+ δ } (cid:0) ⊗ (2 k +2 (cid:96) +1+ δ ) ⊗ ∆ (cid:1) Ξ , or equivalently(Θ − Ξ) ⊗ q + q − Λ { k +2 (cid:96) +3+ δ } (cid:0) ⊗ (2 k +2 (cid:96) +1+ δ ) ⊗ ∆ (cid:1) (Θ − Ξ) . (4.24)We want to show that Θ = Ξ.Let us start by writing Θ − Ξ explicitly as a tensor product, i.e.,Θ − Ξ = (cid:88) i ∈ I a ( i )1 ⊗ · · · ⊗ a ( i )2 k +2 (cid:96) +2+ δ , for certain a ( i ) j ∈ U q ( sl ) and some finite index set I , and where we have grouped the elementsin such a way that the set (cid:8) a ( i )1 ⊗ · · · ⊗ a ( i )2 k +2 (cid:96) +1+ δ : i ∈ I (cid:9) (4.25)is linearly independent. The equality (4.24) then asserts (cid:88) i ∈ I a ( i )1 ⊗ · · · ⊗ a ( i )2 k +2 (cid:96) +1+ δ ⊗ (cid:18) a ( i )2 k +2 (cid:96) +2+ δ ⊗ − q + q − (1 ⊗ Λ)∆ (cid:0) a ( i )2 k +2 (cid:96) +2+ δ (cid:1)(cid:19) = 0 , which by our convention (4.25) implies a ( i )2 k +2 (cid:96) +2+ δ ⊗ q + q − (1 ⊗ Λ)∆ (cid:0) a ( i )2 k +2 (cid:96) +2+ δ (cid:1) for every i ∈ I . But this now implies that every a ( i )2 k +2 (cid:96) +2+ δ vanishes, since if not, then∆ (cid:0) a ( i )2 k +2 (cid:96) +2+ δ (cid:1) = (cid:80) j ∈ J b ( j )1 ⊗ b ( j )2 is nontrivial and must be such that a ( i )2 k +2 (cid:96) +2+ δ ⊗ q + q − (cid:88) j ∈ J b ( j )1 ⊗ Λ b ( j )2 . But by comparing degrees in the generators E and F , it is clear that Λ b ( j )2 (cid:54) = 1 for every possible b ( j )2 ∈ U q ( sl ), i.e., Λ has no inverse. So we have a ( i )2 k +2 (cid:96) +2+ δ = 0 for every i . Hence we haveshown that indeed Θ = Ξ. This concludes the proof. (cid:4) The two preceding proofs manifest the general strategy that can be used to show Proposi-tion 3.1. Therefore, and for the sake of brevity, we will only concisely mention how one canshow the remaining relations.
Lemma 4.9.
The standard relation ( ∗ ) holds for the sets (C1) . igher Rank Relations for the Askey–Wilson and q -Bannai–Ito Algebra 19 Sketch of the proof .
Use ( ∗ ) for A = { , , . . . , k, k + 2 } , B = { , k + 2 } , as followsfrom (C2), to rewrite Λ { , , ,..., k } in[Λ { , , ,..., k } , Λ { , ,..., k, k +1 } ] q . Use (4.3) to switch the order of the nested q -commutators. Use (C4) with (cid:96) = 0 and (1.4) actedupon with χ = −→ k − (cid:81) (cid:96) =1 (cid:0) τ L ⊗ ⊗ (2 (cid:96) +1) (cid:1)(cid:0) ∆ ⊗ ⊗ (cid:96) (cid:1) to expand further. The remaining q -commutatorscan be worked out from (1.3) and (C2). (cid:4) Lemma 4.10.
The standard relation ( ∗ ) holds for the sets (C5) and (C5 (cid:48) ) . Sketch of the proof .
Use ( ∗ ) for (C3) with (cid:96) + 1 instead of k , acted upon with χ = (cid:0) τ L ⊗ ⊗ (2 k +2 (cid:96) + δ ) (cid:1) · · · (cid:0) τ L ⊗ ⊗ (2 (cid:96) +2) (cid:1) , to rewrite Λ B (cid:48) in [Λ A (cid:48) , Λ B (cid:48) ] q with A (cid:48) = { , , . . . , k, k + 1 + δ, k + 3 + δ, . . . , k + 2 (cid:96) + 1 + δ } ,B (cid:48) = { k + 1 + δ, k + 3 + δ, . . . , k + 2 (cid:96) + 1 + δ, k + 2 (cid:96) + 2 + δ } , where δ ∈ { , } . Use (4.2) and expand the inner q -commutator by (C2) acted upon with χ = (cid:0) ⊗ (2 k +2 (cid:96) − δ ) ⊗ τ R (cid:1)(cid:0) ⊗ (2 k +2 (cid:96) − δ ) ⊗ ∆ (cid:1) · · ·× (cid:0) ⊗ (2 k +1+ δ ) ⊗ τ R (cid:1)(cid:0) ⊗ (2 k + δ ) ⊗ ∆ (cid:1)(cid:0) ⊗ k ⊗ τ R (cid:1) δ . Apply the relation ( ∗ ) for the sets A = { , k + 2 (cid:96) + 2 + δ } ,B = { , , , . . . , k } ; A = { , k + 2 (cid:96) + 2 + δ } ,B = { , , , . . . , k, k + 1 + δ, k + 3 + δ, . . . , k + 2 (cid:96) + 1 + δ } , as follows from (C3) by χ = (cid:0) ⊗ (2 k +2 (cid:96) + δ ) ⊗ τ R (cid:1) · · · (cid:0) ⊗ k ⊗ τ R (cid:1) respectively from (C3) with k + (cid:96) + δ instead of k , acted upon with χ = (cid:0) ⊗ (2 k − ⊗ ∆ ⊗ ⊗ (2 (cid:96) +1) (cid:1) − δ . In our expression for[Λ A (cid:48) , Λ B (cid:48) ] q , there is now one term left containing a q -commutator, namely1 q + q − Λ { } [Λ { , ,..., k, k +1+ δ, k +3+ δ,..., k +2 (cid:96) +1+ δ } , Λ { , k +1+ δ, k +3+ δ,..., k +2 (cid:96) +1+ δ, k +2 (cid:96) +2+ δ } ] q . (4.26)On the other hand, we already know how to expand [Λ A (cid:48) , Λ B (cid:48) ] q from (C1) with (cid:96) + 1 insteadof k , acted upon with χ = (cid:0) τ L ⊗ ⊗ (2 k +2 (cid:96) − δ ) (cid:1)(cid:0) ∆ ⊗ ⊗ (2 k +2 (cid:96) − δ ) (cid:1) · · ·× (cid:0) τ L ⊗ ⊗ (2 (cid:96) +3+ δ ) (cid:1)(cid:0) ∆ ⊗ ⊗ (2 (cid:96) +2+ δ ) (cid:1)(cid:0) τ L ⊗ ⊗ (2 (cid:96) +2) (cid:1) δ , and adding a factor 1 ⊗ in front. This leads to the sought expression for the remaining q -commutator in (4.26). (cid:4) To show the final cases (C6) and (C6 (cid:48) ), we will need another commutation relation.0 H. De Clercq
Lemma 4.11.
For any k ∈ N one has [Λ { , k +1 } , Λ { , , ,..., k, k +1 } ] = 0 . Proof .
By induction on k , the case k = 1 being obvious from a direct calculation. Suppose theclaim has been proven for k −
1. Since ( ∗ ) holds for (C5), with k − k and with (cid:96) = 0,we have by χ = 1 ⊗ (2 k − ⊗ ∆Λ { , , ,..., k, k +1 } = [Λ { , ,..., k − , k − } , Λ { , k − , k, k +1 } ] q q − − q + Λ { k − } Λ { , , ,..., k − , k − , k, k +1 } + Λ { , ,..., k − } Λ { , k, k +1 } q + q − . It suffices now to show that each of the generators in the right-hand side commutes withΛ { , k +1 } . For Λ { , ,..., k − , k − } and Λ { , ,..., k − } this follows from Lemma 4.5 with k − k , upon applying χ = (cid:0) ⊗ (2 k − ⊗ τ R (cid:1)(cid:0) ⊗ (2 k − ⊗ ∆ ⊗ (cid:1) and χ = (cid:0) ⊗ (2 k − ⊗ τ R (cid:1)(cid:0) ⊗ (2 k − ⊗ τ R (cid:1) respectively. For Λ { , , ,..., k − , k − , k, k +1 } we apply χ = (cid:0) ⊗ (2 k − ⊗ ∆ ⊗ ⊗ (cid:1)(cid:0) ⊗ (2 k − ⊗ ∆ ⊗ (cid:1) to the induction hypothesis. The remaining nontrivial commutation relations follow from[Λ { , } , Λ { , , } ] = 0 upon applying respectively χ = (cid:0) τ L ⊗ ⊗ (2 k − (cid:1) · · · (cid:0) τ L ⊗ ⊗ (cid:1)(cid:0) τ L ⊗ ⊗ (cid:1) (1 ⊗ ∆ ⊗ χ = (cid:0) τ L ⊗ ⊗ (2 k − (cid:1) · · · (cid:0) τ L ⊗ ⊗ (cid:1)(cid:0) τ L ⊗ ⊗ (cid:1) . (cid:4) Corollary 4.1. Λ { , k } commutes with Λ { , , ,..., k } and Λ { , , ,..., k − , k } . Proof .
Act on the relation of Lemma 4.11, with k − k , with χ = 1 ⊗ (2 k − ⊗ τ R and χ = τ L ⊗ ⊗ (2 k − respectively. (cid:4) Lemma 4.12.
The standard relation ( ∗ ) holds for the sets (C6) and (C6 (cid:48) ) , i.e., for A = { , k + 1 + δ, k + 3 + δ, . . . , k + 2 (cid:96) + 1 + δ, k + 2 (cid:96) + 2 + δ } ,B = { , , , . . . , k, k + 2 (cid:96) + 2 + δ } (4.27) with k, (cid:96) ∈ N and δ ∈ { , } . Sketch of the proof .
By induction on k . Already the induction basis k = 1, i.e., to prove ( ∗ )for A = { , δ, δ, . . . , (cid:96) + 3 + δ, (cid:96) + 4 + δ } , B = { , , (cid:96) + 4 + δ } , (4.28)is difficult and needs to be shown by induction on (cid:96) , the case (cid:96) = 0 being trivial. Suppose thusthat the claim holds for (cid:96) −
1, i.e., the relation ( ∗ ) holds for A = { , δ, δ, . . . , (cid:96) + 1 + δ, (cid:96) + 2 + δ } , B = { , , (cid:96) + 2 + δ } . (4.29)We will find an expression for[Λ { , δ, δ,..., (cid:96) +3+ δ, (cid:96) +4+ δ } , Λ { , , (cid:96) +4+ δ } ] q . (4.30)Rewrite Λ { , δ, δ,..., (cid:96) +3+ δ, (cid:96) +4+ δ } in (4.30) by (C4) with k = 1 and (cid:96) = 0, acted upon by χ = (cid:0) τ L ⊗ ⊗ (2 (cid:96) +3) (cid:1) δ (cid:0) τ L ⊗ ⊗ (2 (cid:96) +2) (cid:1)(cid:0) ∆ ⊗ ⊗ (2 (cid:96) +1) (cid:1) · · ·× (cid:0) τ L ⊗ ⊗ (cid:1)(cid:0) ∆ ⊗ ⊗ (cid:1)(cid:0) τ L ⊗ ⊗ (cid:1)(cid:0) ∆ ⊗ ⊗ (cid:1) . igher Rank Relations for the Askey–Wilson and q -Bannai–Ito Algebra 21Use (4.3) and expand further by ( ∗ ) for A = { , δ, δ, . . . , (cid:96) + 1 + δ, (cid:96) + 2 + δ, (cid:96) + 4 + δ } ,B = { , , (cid:96) + 4 + δ } ; A = { , δ, δ, . . . , (cid:96) + 1 + δ, (cid:96) + 2 + δ, (cid:96) + 3 + δ, (cid:96) + 4 + δ } ,B = { , , (cid:96) + 4 + δ } ; A = { , δ, δ, . . . , (cid:96) + 1 + δ, (cid:96) + 4 + δ } ,B = { , , (cid:96) + 4 + δ } , as follows from the induction hypothesis (4.29) by respectively χ = (cid:0) ⊗ (2 (cid:96) + δ ) ⊗ ∆ ⊗ ⊗ (cid:1)(cid:0) ⊗ (2 (cid:96) +1+ δ ) ⊗ τ R (cid:1) ,χ = (cid:0) ⊗ (2 (cid:96) + δ ) ⊗ ∆ ⊗ ⊗ (cid:1)(cid:0) ⊗ (2 (cid:96) + δ ) ⊗ ∆ ⊗ (cid:1) ,χ = (cid:0) ⊗ (2 (cid:96) +2+ δ ) ⊗ τ R (cid:1)(cid:0) ⊗ (2 (cid:96) +1+ δ ) ⊗ τ R (cid:1) . Use the relation [Λ { , (cid:96) +4+ δ } , Λ { , , δ, δ,..., (cid:96) +1+ δ, (cid:96) +2+ δ, (cid:96) +4+ δ } ] = 0, as follows from Lem-ma 4.11 with (cid:96) + δ instead of k , after applying χ = (cid:0) ⊗ ∆ ⊗ ⊗ (2 (cid:96) +1) (cid:1) − δ (cid:0) ⊗ (2 (cid:96) − δ ) ⊗ ∆ ⊗ ⊗ (cid:1)(cid:0) ⊗ (2 (cid:96) +2 δ ) ⊗ τ R (cid:1) . The remaining q -commutators can now be expanded using Lemma 4.4 and (C4) with k = 1 and (cid:96) = 0, acted upon by χ = (cid:0) ∆ ⊗ ⊗ (2 (cid:96) +2+ δ ) (cid:1)(cid:0) τ L ⊗ ⊗ (2 (cid:96) +2) (cid:1) δ (cid:0) ∆ ⊗ ⊗ (2 (cid:96) +1) (cid:1)(cid:0) τ L ⊗ ⊗ (cid:96) (cid:1)(cid:0) ∆ ⊗ ⊗ (2 (cid:96) − (cid:1) · · ·× (cid:0) τ L ⊗ ⊗ (cid:1)(cid:0) ∆ ⊗ ⊗ (cid:1) . This leads to ( ∗ ) for (4.28), which will serve as the basis for the induction on k we perform inour global proof, i.e., to show ( ∗ ) for (4.27).Suppose now the claim holds for k −
1, i.e., the relation ( ∗ ) holds for A = { , k − δ, k + 1 + δ, . . . , k + 2 (cid:96) − δ, k + 2 (cid:96) + δ } ,B = { , , , . . . , k − , k + 2 (cid:96) + δ } , (4.31)for arbitrary (cid:96) ∈ N . We will find an expression for[Λ { , k +1+ δ, k +3+ δ,..., k +2 (cid:96) +1+ δ, k +2 (cid:96) +2+ δ } , Λ { , , ,..., k, k +2 (cid:96) +2+ δ } ] q . (4.32)Use (C5) with k = 1 and (cid:96) = 0, acted upon with χ = −−−−→ k +2 (cid:96) + δ (cid:89) m =2 k +1 (cid:0) ⊗ m ⊗ τ R (cid:1) −−→ k (cid:89) m =2 (cid:0) ⊗ m ⊗ τ R (cid:1)(cid:0) ⊗ (2 m − ⊗ ∆ (cid:1) , to rewrite Λ { , , ,..., k, k +2 (cid:96) +2+ δ } in (4.32) and apply (4.2). Expand further by ( ∗ ) for A = { , k + 1 + δ, k + 3 + δ, . . . , k + 2 (cid:96) + 1 + δ, k + 2 (cid:96) + 2 + δ } and B = { , , , , . . . , k − , k, k + 2 (cid:96) + 2 + δ } ,B = { , , , , , . . . , k − , k, k + 2 (cid:96) + 2 + δ } ,B = { , , , . . . , k − , k, k + 2 (cid:96) + 2 + δ } , χ = (cid:0) τ L ⊗ ⊗ (2 k +2 (cid:96) + δ ) (cid:1)(cid:0) ⊗ ∆ ⊗ ⊗ (2 k +2 (cid:96) − δ ) (cid:1) ,χ = (cid:0) ⊗ ∆ ⊗ ⊗ (2 k +2 (cid:96) − δ ) (cid:1)(cid:0) ⊗ ∆ ⊗ ⊗ (2 k +2 (cid:96) − δ ) (cid:1) ,χ = (cid:0) τ L ⊗ ⊗ (2 k +2 (cid:96) + δ ) (cid:1)(cid:0) τ L ⊗ ⊗ (2 k +2 (cid:96) − δ ) (cid:1) . Apply the commutation relation[Λ { , k +2 (cid:96) +2+ δ } , Λ { , , , ,..., k, k +1+ δ, k +3+ δ,..., k +2 (cid:96) +1+ δ, k +2 (cid:96) +2+ δ } ] = 0 , which follows from Lemma 4.11. The remaining q -commutators can now be expanded using Lem-ma 4.4 and (C5) with k = 1 and (cid:96) = 0. This leads to the anticipated expression for (4.32). (cid:4) In this section we derive several criteria on the sets A and B for the elements Λ A and Λ B tocommute. Eventually, it will be our aim to show that it suffices that B ⊆ A in order to have[Λ A , Λ B ] = 0. This result, stated in Theorem 3.1, will be proven in Section 6 and its proof willrely on the results of the present section. Lemma 5.1.
The element Λ { , , ,..., k − } commutes with Λ [1;2 k ] and Λ [1;2 k − , and Λ { , , ,..., k } commutes with Λ [1;2 k ] and Λ [1;2 k +1] . Proof .
We will only show the first relation, the others follow in complete analogy. We proceedby induction on k , the case k = 1 being trivial. Suppose hence that the claim holds for k − A commutes with Λ B for A = { , , , . . . , k − } , B = [1; 2 k − . (5.1)By Lemma 4.4 we may writeΛ { , , ,..., k − } = [Λ { , , ,..., k − , k − } , Λ { k − , k − } ] q q − − q + Λ { k − } Λ { , , ,..., k − , k − , k − } + Λ { k − } Λ { , , ,..., k − } q + q − . Hence it suffices to show that Λ [1;2 k ] commutes with each of the terms in the right-hand side.For Λ { , , ,..., k − , k − } , Λ { , , ,..., k − , k − , k − } and Λ { , , ,..., k − } this follows from the inductionhypothesis (5.1) by χ = (cid:0) ⊗ (2 k − ⊗ ∆ (cid:1)(cid:0) ⊗ (2 k − ⊗ ∆ ⊗ (cid:1) , χ = (cid:0) ⊗ (2 k − ⊗ ∆ ⊗ (cid:1)(cid:0) ⊗ (2 k − ⊗ ∆ ⊗
1) and χ = (cid:0) ⊗ (2 k − ⊗ ∆ (cid:1)(cid:0) ⊗ (2 k − ⊗ ∆ (cid:1) respectively. For Λ { k − , k − } this follows from[Λ { } , Λ { , , } ] = 0 acted upon with 1 ⊗ ∆ ⊗ ⊗ ⊗ n . (cid:4) Another useful commutation relation relies on Lemma 4.12. The proof is similar to the oneabove, we will hence just sketch it.
Lemma 5.2.
The element Λ { , ,..., k } commutes with Λ { , , ,..., k, k +1 } . Sketch of the proof .
By induction on k . Rewrite Λ { , ,..., k } by ( ∗ ) for (C6 (cid:48) ), with k − k and (cid:96) = 0. Check that Λ { , , ,..., k, k +1 } commutes with all generators in the newexpression. This requires us to use Lemma 4.11, the first statement of Corollary 4.1 acted uponwith 1 ⊗ (2 k − ⊗ ∆, the induction hypothesis acted upon with (cid:0) ⊗ (2 k − ⊗ ∆ (cid:1)(cid:0) ⊗ (2 k − ⊗ τ R (cid:1) andthe relation [Λ { , } , Λ { , , } ] = 0 acted upon with iterations of (cid:0) ∆ ⊗ ⊗ ( n +1) (cid:1)(cid:0) τ L ⊗ ⊗ n (cid:1) . (cid:4) igher Rank Relations for the Askey–Wilson and q -Bannai–Ito Algebra 23In the following commutation relations, the indexing subsets will no longer be defined byintegers k and (cid:96) , but rather by more general conditions. Lemma 5.3.
Let A be a set of consecutive integers and B ⊆ A . Then Λ A and Λ B commute. Proof .
Without loss of generality, we may assume A = [1; n ] and write B as B = [ i ; j ] ∪ · · · ∪ [ i k ; j k ] . The claim follows upon applying −−−−−→ n − (cid:89) (cid:96) = n − i +1 (cid:0) ∆ ⊗ ⊗ (cid:96) (cid:1) − δ i , −−−−−−−−−−→ n − i − δ i , (cid:89) (cid:96) = j k − i +2 − δ i , (cid:0) ⊗ (cid:96) ⊗ ∆ (cid:1) − δ jk,n χ (cid:48) , with χ (cid:48) = ←−−−−−− k − − δ i , (cid:89) m =1 − δ i , −−−−−−−−−−→ β n, i , j +1 − δ jk,n (cid:89) (cid:96) = α n, i , j +1 − δ jk,n (cid:0) ⊗ m ⊗ ∆ ⊗ ⊗ (cid:96) (cid:1) to the commutation relation (∆) for A = { − δ i , , − δ i , , . . . , k − δ i , } , B = [1; 2 k + 1 − δ i , − δ j k ,n ] , which follows from Lemma 5.1. (cid:4) It can easily be checked explicitly that one has[Λ { , , } , Λ { , } ] = 0 , [Λ { , , } , Λ { , } ] = 0 , [Λ { , , , } , Λ { , } ] = 0 , [Λ { , , } , Λ { , } ] = 0 , [Λ { , , , } , Λ { , } ] = 0 , [Λ { , , , } , Λ { , } ] = 0 , [Λ { , , , , } , Λ { , } ] = 0 , [Λ { , , } , Λ { , } ] = 0 , [Λ { , , , } , Λ { , } ] = 0 , [Λ { , , , } , Λ { , } ] = 0 , [Λ { , , , , } , Λ { , } ] = 0 , [Λ { , , , } , Λ { , } ] = 0 , [Λ { , , , , } , Λ { , } ] = 0 , [Λ { , , , , } , Λ { , } ] = 0 , [Λ { , , , , , } , Λ { , } ] = 0 . (5.2)These relations are needed in order to show the following result. Lemma 5.4.
Let A = [1; j ] ∪ [ i ; j ] and B be such that B ⊆ A and | B | = 2 , then Λ A and Λ B commute. Sketch of the proof . If B ⊆ [1; j ], then by acting with morphisms 1 ⊗ (cid:96) ⊗ τ R and 1 ⊗ m ⊗ ∆,the claim follows from [Λ [1; j +1] , Λ B ] = 0, as asserted by Lemma 5.3. Similarly if B ⊆ [ i ; j ].Now suppose B (cid:42) [1; j ] and B (cid:42) [ i ; j ], then B ∩ [1; j ] consists of a single element n and B ∩ [ i ; j ] of a single element n . One can distinguish 16 cases: each of the 4 cases n = 1 = j , n = 1 (cid:54) = j , n = j (cid:54) = 1, j (cid:54) = n (cid:54) = 1 can be combined with each of the 4 cases determinedsimilarly by the mutual equality of n , j and i . Each of those cases follows from one of therelations in (5.2) upon applying ∆ on suitable tensor product positions. (cid:4) The following lemma aims to remove part of the restrictions on the set B . Lemma 5.5.
Let A = [1; j ] ∪ [ i ; j ] , and B be such that B ⊆ A and | B ∩ [ i ; j ] | = 1 , then Λ A and Λ B commute. Proof .
By induction on k = | B ∩ [1; j ] | , the case k = 0 being trivial by Lemma 4.2 and k = 1following from Lemma 5.4. So let k ≥ B (cid:48) ⊂ A with | B (cid:48) ∩ [1; j ] | strictly less than k .If B ∩ [1; j ] is a set of consecutive integers, the statement follows from Lemma 5.4 uponapplying ∆ on suitable positions as in Proposition 2.4. If not, then B ∩ [1; j ] contains atleast one hole, say between the elements x and x . Let us write B = { b ∈ B : b < x } and B = { b ∈ B : b > x } . By Lemma 4.4 we may writeΛ B = [Λ B ∪ [ x ; x − , Λ [ x +1; x ] ∪ B ] q q − − q + Λ [ x +1; x − Λ B ∪ [ x ; x ] ∪ B + Λ B ∪{ x } Λ { x }∪ B q + q − (5.3)and hence it suffices to show that Λ A commutes with each of the terms in the right-hand side.For Λ B ∪ [ x ; x − , Λ [ x +1; x − and Λ B ∪{ x } this follows from Lemma 5.3 with A = [1; j + 1],upon applying repeatedly 1 ⊗ m ⊗ τ R and 1 ⊗ (cid:96) ⊗ ∆. For Λ [ x +1; x ] ∪ B , Λ B ∪ [ x ; x ] ∪ B and Λ { x }∪ B this follows from the induction hypothesis. This concludes the proof. (cid:4) We can now also remove the constraint on | B ∩ [ i ; j ] | . The proof is completely similar tothe previous one, we will hence only sketch it. Proposition 5.1.
Let A = [1; j ] ∪ [ i ; j ] and B be such that B ⊆ A , then Λ A commuteswith Λ B . Sketch of the proof . If B ∩ [1; j ] is empty, the result follows from Lemma 5.3. If it is not,then we will proceed by induction on k = | B ∩ [ i ; j ] | . The case k = 0 follows from Lemma 5.3and k = 1 from Lemma 5.5. Let hence k ≥ | B ∩ [ i ; j ] | strictly less than k . We again distinguish between B ∩ [ i ; j ] without holes, inwhich case Lemma 5.5 suffices, and B ∩ [ i ; j ] containing holes, in which case we can rewrite Λ B using (5.3) and apply Lemma 5.3 and the induction hypothesis. (cid:4) Let us now take a look at the case where the set A consists of more than just 2 discreteintervals. Lemma 5.6.
Let A = [1; j ] ∪ [ i ; j ] ∪ · · · ∪ [ i k − ; j k − ] ∪ [ i k ; j k ] , with k ≥ . Then Λ A commuteswith Λ B , where B is any of the sets B = [ i (cid:48) ; j ] ∪ [ i ; j ] ∪ · · · ∪ [ i k − ; j k − ] ∪ [ i k ; j (cid:48) k ] , B = [ i (cid:48) ; j ] ∪ [ i ; j ] ∪ · · · ∪ [ i k − ; j k − ] ,B = [ i ; j ] ∪ · · · ∪ [ i k − ; j k − ] ∪ [ i k ; j (cid:48) k ] , B = [ i ; j ] ∪ · · · ∪ [ i k − ; j k − ] , where ≤ i (cid:48) ≤ j and i k ≤ j (cid:48) k ≤ j k . Proof .
Let us start with the first claim. If i (cid:48) = 1 or j (cid:48) k = j k , this follows from Lemma 4.3. Sosuppose 1 < i (cid:48) , j (cid:48) k < j k . Then the statement follows upon applying χ = −−−−−−→ j k − (cid:89) (cid:96) = j k − i (cid:48) +1 (cid:0) ∆ ⊗ ⊗ (cid:96) (cid:1) −−−−→ j k − i (cid:48) − (cid:89) (cid:96) = j k − j (cid:0) ⊗ ∆ ⊗ ⊗ (cid:96) (cid:1) ←−− k − (cid:89) n =1 −−−−→ β n, i , j (cid:89) (cid:96) = α n, i , j (cid:0) ⊗ ( n +1) ⊗ ∆ ⊗ ⊗ (cid:96) (cid:1) × −−−−→ j k − i k − (cid:89) (cid:96) = j k − j (cid:48) k (cid:0) ⊗ (2 k − ⊗ ∆ ⊗ ⊗ (cid:96) (cid:1) −−−−→ j k − j (cid:48) k − (cid:89) (cid:96) =0 (cid:0) ⊗ k ⊗ ∆ ⊗ ⊗ (cid:96) (cid:1) to the statement of Lemma 5.2, as asserted by Proposition 2.4. The second claim follows fromthe first with A replaced by [1; j ] ∪ [ i ; j ] ∪ · · · ∪ [ i k − ; j k − + 1], acted upon by iterations of1 ⊗ (cid:96) ⊗ τ R and 1 ⊗ m ⊗ ∆. Similarly for the other statements. (cid:4) igher Rank Relations for the Askey–Wilson and q -Bannai–Ito Algebra 25For any m ∈ N one can define the following operators: χ m = ←− m (cid:89) n =1 −−−−−−→ j k − i n − δ n, (cid:89) (cid:96) = j k − j n (cid:0) ∆ ⊗ ⊗ (cid:96) (cid:1) −−−−−−−→ j k − j n − (cid:89) (cid:96) = j k − i n +1 +1 (cid:0) τ L ⊗ ⊗ (cid:96) (cid:1) , (5.4) (cid:101) χ m = −−→ k (cid:89) n = m −−−−−→ j n − − δ n,k (cid:89) (cid:96) = i n − (cid:0) ⊗ (cid:96) ⊗ ∆ (cid:1) −−−→ i n − (cid:89) (cid:96) = j n − (cid:0) ⊗ (cid:96) ⊗ τ R (cid:1) . (5.5)These will be of use to show the following commutation relation. Lemma 5.7.
Let A = [1; j ] ∪ [ i ; j ] ∪ · · · ∪ [ i k − ; j k − ] ∪ [ i k ; j k ] and let B be any of the sets B = [ i (cid:48) ; j ] ∪ [ i ; j ] ∪ · · · ∪ [ i k − ; j k − ] ∪ B k , B = [ i ; j ] ∪ · · · ∪ [ i k − ; j k − ] ∪ B k , where k ≥ , ≤ i (cid:48) ≤ j and where B k ⊆ [ i k ; j k ] . Then Λ A and Λ B commute. Proof .
We will start with the first claim. If B k is empty or of the form [ i k ; j (cid:48) k ] with i k ≤ j (cid:48) k ≤ j k ,the statement follows from Lemma 5.6. So suppose B k is nonempty and not of that particularform. We will prove the special case i (cid:48) = j = 2, the general case then follows upon applying ∆on tensor product positions 1 and 2. Let x = min([ i k ; j k ] \ B k ). Define the sets B = { b ∈ B : b < x } = { } ∪ [ i ; j ] ∪ · · · ∪ [ i k − ; j k − ] ∪ [ i k ; x − ,B = { b ∈ B k : b > x } , where we interpret [ i k ; x −
1] as the empty set in case x = i k . Note that both sets are nonempty,by the assumptions on B k . By Lemma 4.4 we haveΛ B = [Λ B ∪{ x } , Λ { x }∪ B ] q q − − q + Λ { x } Λ B ∪{ x } + Λ B Λ B q + q − . (5.6)Both Λ B and Λ B ∪{ x } commute with Λ A , as follows from Lemma 5.6 and the form of B .Moreover, Lemma 5.3 asserts[Λ [1; j k − i k +2] , Λ B − ( i k − ] = 0 , [Λ [1; j k − i k +2] , Λ ( { x }∪ B ) − ( i k − ] = 0 . Acted upon with χ k − , defined in (5.4), this implies Λ A commutes with Λ B and Λ { x }∪ B .Hence it follows from (5.6) that[Λ A , Λ B ] = 1 q + q − Λ { x } [Λ A , Λ B ∪{ x } ] . We now repeat our reasoning. Either B k ∪{ x } is of the form [ i k ; j (cid:48) k ], in which case the statementfollows from Lemma 5.6. If not, then upon defining x = min([ i k ; j k ] \ ( B k ∪ { x } )), we find bythe same arguments[Λ A , Λ B ∪{ x } ] = 1 q + q − Λ { x } [Λ A , Λ B ∪{ x ,x } ] . If we continue this process, then at some point the set B k ∪ { x , x , . . . , x m } will inevitably beof the form [ i k ; j (cid:48) k ], with j (cid:48) k = max( B ), namely when we have filled up all the holes in the set B k .At this point we have[Λ A , Λ B ] = 1 (cid:0) q + q − (cid:1) m (cid:32) m (cid:89) (cid:96) =1 Λ { x (cid:96) } (cid:33) [Λ A , Λ B ∪{ x ,...,x m } ] = 0 , where the last step uses Lemma 5.6. This shows the claim for the first given form of B .6 H. De ClercqFor the second form, our first claim asserts[Λ ([ i − j ] ∪···∪ [ i k − ; j k − ] ∪ [ i k ; j k ]) − ( i − , Λ ([ i ; j ] ∪···∪ [ i k − ; j k − ] ∪ B k ) − ( i − ] = 0 , which yields the anticipated result by the left extension process. (cid:4) We can now also replace [ i (cid:48) ; j ] in the previous lemma by an arbitrary set. The proof will becompletely parallel to the previous one. Proposition 5.2.
Let A = [1; j ] ∪ [ i ; j ] ∪ · · · ∪ [ i k − ; j k − ] ∪ [ i k ; j k ] , with k ≥ , and let B besuch that B ⊆ A and [ i ; j ] ∪ · · · ∪ [ i k − ; j k − ] ⊆ B , then Λ A commutes with Λ B . Sketch of the proof . If B ∩ [1; j ] is empty or of the form [ i (cid:48) ; j ], the claim follows fromLemma 5.7. If not, then with x = max([1; j ] \ B ), B = { b ∈ B : b < x } and B = { b ∈ B : b > x } = [ x +1; j ] ∪ [ i ; j ] ∪· · ·∪ [ i k − ; j k − ] ∪ ( B ∩ [ i k ; j k ]), we have again the relation (5.6).Each of the terms in the right-hand side of (5.6) commutes with Λ A , by Lemmas 5.3 and 5.7,except for Λ B ∪{ x } . Defining recursively x i = max([1; j ] \ ( B ∪ { x , . . . , x i − } )), we find[Λ A , Λ B ] = 1 q + q − Λ { x } [Λ A , Λ B ∪{ x } ] = 1 (cid:0) q + q − (cid:1) Λ { x } Λ { x } [Λ A , Λ B ∪{ x ,x } ] = · · · , which eventually becomes zero, since at some point the set ( B ∪ { x , . . . , x m } ) ∩ [1; j ] willinevitably be of the form [ i (cid:48) ; j ], with i (cid:48) = min( B ), such that Lemma 5.7 will be applicable. (cid:4) Proof .
Without loss of generality, we may assume min( A ) = 1 and thus we can write A =[1; j ] ∪ [ i ; j ] ∪ · · · ∪ [ i k ; j k ]. For ease of notation, we will write A (cid:96) for the discrete interval [ i (cid:96) ; j (cid:96) ].We will proceed by induction on k .For k = 1 the statement follows from Lemma 5.3. For k = 2 we may invoke Proposition 5.1.Suppose hence that k ≥ A (cid:48) consisting of strictly lessthan k discrete intervals, and for any set B (cid:48) contained in A (cid:48) . We distinguish four cases. Case 1: A ∪ · · · ∪ A k − ⊆ B .This case follows immediately from Proposition 5.2. Case 2: B ∩ A k = ∅ .In this case the induction hypothesis asserts [Λ A ∪···∪ A k − ∪{ j k − +1 } , Λ B ] = 0, which impliesour claim upon acting repeatedly with 1 ⊗ (cid:96) ⊗ τ R and 1 ⊗ m ⊗ ∆. Case 3: B ∩ A = ∅ .Similarly, using the induction hypothesis and the left extension process. Case 4: A ∪ · · · ∪ A k − (cid:42) B , B ∩ A (cid:54) = ∅ and B ∩ A k (cid:54) = ∅ .Let x ∈ A \ B be such that x ∈ A i with i ∈ { , . . . , k − } . Let us define the sets B = { b ∈ B : b < x } and B = { b ∈ B : b > x } . Then it follows from Lemma 4.4 thatΛ B = [Λ B ∪{ x } , Λ { x }∪ B ] q q − − q + Λ { x } Λ B ∪{ x } + Λ B Λ B q + q − . (6.1)Note that none of the indexing sets above is empty, by our assumptions on B . Observe that B and B ∪ { x } are contained in A ∪ · · · ∪ A i ∪ { j i + 1 } , which contains strictly fewer discreteintervals than k . Hence the induction hypothesis asserts[Λ B , Λ A ∪···∪ A i ∪{ j i +1 } ] = 0 , [Λ B ∪{ x } , Λ A ∪···∪ A i ∪{ j i +1 } ] = 0 , igher Rank Relations for the Askey–Wilson and q -Bannai–Ito Algebra 27which by (cid:101) χ i +1 , as defined in (5.5), asserts that Λ B and Λ B ∪{ x } commute with Λ A . A similarreasoning using the induction hypothesis and the morphism χ i − , defined in (5.4), shows that Λ A commutes with Λ B and Λ { x }∪ B . The expression (6.1) now implies[Λ A , Λ B ] = 1 q + q − Λ { x } [Λ A , Λ B ∪{ x } ] . We may now repeat this reasoning. Either B ∪ { x } contains A ∪ · · · ∪ A k − , in whichcase [Λ A , Λ B ∪{ x } ] = 0 by Proposition 5.2. If not, then as established before A \ B contains anelement x contained in A j for a certain j ∈ { , . . . , k − } , and as before this implies that[Λ A , Λ B ∪{ x } ] = 1 q + q − Λ { x } [Λ A , Λ B ∪{ x ,x } ] . If we continue this process, then at some point the set B ∪ { x , . . . , x m } must inevitably contain A ∪ · · · ∪ A k − . At this point we have[Λ A , Λ B ] = 1 (cid:0) q + q − (cid:1) m (cid:32) m (cid:89) (cid:96) =1 Λ { x (cid:96) } (cid:33) [Λ A , Λ B ∪{ x ,...,x m } ] = 0 , where in the last step we have used Proposition 5.2. This concludes the proof. (cid:4) Proof .
We will start by proving the case A , A , A (cid:54) = ∅ and A = ∅ . It suffices to do thisin the situation where A = { } , min( A ) = 2 and A = { max( A ) + 1 } , the general case thenfollows upon applying suitable morphisms of the form ∆ ⊗ ⊗ n , τ L ⊗ ⊗ n , 1 ⊗ n ⊗ ∆ and 1 ⊗ n ⊗ τ R .Let us write A in the form A = [2; j ] ∪ [ i ; j ] ∪ · · · ∪ [ i k ; j k ] . Cases (3.1), (3.2) and (3.3) then follow from (C1), (C2) and (C3) respectively, by χ = −−−−−−→ j k − (cid:89) (cid:96) = j k − j +1 (cid:0) ⊗ ∆ ⊗ ⊗ (cid:96) (cid:1) ←−− k − (cid:89) n =1 −−−−−→ β n, i , j +1 (cid:89) (cid:96) = α n, i , j +1 (cid:0) ⊗ ( n +1) ⊗ ∆ ⊗ ⊗ (cid:96) (cid:1) . Indeed, it follows from Proposition 2.4, after separating the terms corresponding to n = 0 andapplying coassociativity, thatΛ A = −−−−−−→ j k − (cid:89) (cid:96) = j k − j +1 (cid:0) ⊗ ∆ ⊗ ⊗ (cid:96) (cid:1) (cid:0) ∆ ⊗ ⊗ ( j k − j +1) (cid:1) × ←−− k − (cid:89) n =1 −−−−−→ β n, i , j +1 (cid:89) (cid:96) = α n, i , j +1 (cid:0) ⊗ n ⊗ ∆ ⊗ ⊗ (cid:96) (cid:1) Λ { , , ,..., k − } , where Λ { , , ,..., k − } is considered inside a (2 k )-fold tensor product. The morphism ∆ ⊗ ⊗ ( j k − j +1) can be shifted through the subsequent ones by (2.11), thus acting directly on Λ { , , ,..., k − } by sending it to Λ { , , , ,..., k } . Hence we have Λ A = χ (Λ { , , , ,..., k } ). Similarly, χ sendsΛ { , , ,..., k, k +1 } to Λ B and so on.8 H. De ClercqThe case A = ∅ and A , A , A (cid:54) = ∅ is identical.Now consider the case A = ∅ and A , A , A (cid:54) = ∅ . Then (3.2) and (3.3) follow fromTheorem 3.1 and (2.8). To show (3.1), it again suffices to prove [Λ A , Λ B ] = 0, by (2.8), and totake A = { } , min( A ) = 2 and A = { max( A ) + 1 } . Let us write A in the form A = [2; j ] ∪ [ i ; j ] ∪ · · · ∪ [ i k ; j k ] . By χ = ←−− k − (cid:89) n =0 −−−−→ β n, i , j (cid:89) (cid:96) = α n, i , j (cid:0) ⊗ ( n +1) ⊗ ∆ ⊗ ⊗ ( (cid:96) +1) (cid:1) with i = 2, this follows from Lemma 4.5.The case A = ∅ and A , A , A (cid:54) = ∅ is identical.If two or more of the A i are empty, then again the statements follow from Theorem 3.1and (2.8).Finally, suppose none of the A i is empty. Again, it suffices to consider the special case where A = { } , min( A ) = 2 and A = { max( A ) + 1 } . We will write A and A as disjoint unionsof discrete intervals. If min( A ) = max( A ) + 1, then A = [2; j ] ∪ [ i ; j ] ∪ · · · ∪ [ i k ; j k ] ,A = [ j k + 1; j k +1 ] ∪ [ i k +2 ; j k +2 ] · · · ∪ [ i k + (cid:96) +1 ; j k + (cid:96) +1 ] . Cases (3.1), (3.2) and (3.3) then follow from (C4), (C5) and (C6) respectively, by χ = ←−− k − (cid:89) n =0 −−−−−→ β n, i , j +1 (cid:89) (cid:96) = α n, i , j +1 (cid:0) ⊗ ( n +1) ⊗ ∆ ⊗ ⊗ (cid:96) (cid:1) −−−−−−−−−−→ j k + (cid:96) +1 − j k − (cid:89) (cid:96) = j k + (cid:96) +1 − j k +1 +1 (cid:0) ⊗ k ⊗ ∆ ⊗ ⊗ (cid:96) (cid:1) × ←−−−− k +2 (cid:96) (cid:89) n =2 k +1 −−−−−→ β n, i , j +1 (cid:89) (cid:96) = α n, i , j +1 (cid:0) ⊗ n ⊗ ∆ ⊗ ⊗ (cid:96) (cid:1) , where i = (2 , i , . . . , i k , j k + 1 , i k +2 , . . . , i k + (cid:96) +1 ) and j = ( j , . . . , j k + (cid:96) +1 ).If min( A ) > max( A ) + 1, then we have A = [2; j ] ∪ [ i ; j ] ∪ · · · ∪ [ i k ; j k ] , A = [ i k +1 ; j k +1 ] ∪ [ i k +2 ; j k +2 ] · · · ∪ [ i k + (cid:96) +1 ; j k + (cid:96) +1 ]and by χ = ←−−− k +2 (cid:96) (cid:89) n =0 −−−−−→ β n, i , j +1 (cid:89) (cid:96) = α n, i , j +1 (cid:0) ⊗ ( n +1) ⊗ ∆ ⊗ ⊗ (cid:96) (cid:1) , with i = (2 , i , . . . , i k + (cid:96) +1 ) and j = ( j , . . . , j k + (cid:96) +1 ), the cases (3.1), (3.2) and (3.3) follow from(C4 (cid:48) ), (C5 (cid:48) ) and (C6 (cid:48) ) respectively. (cid:4) q -Bannai–Ito relations Each of the results and proofs outlined in this paper carries over to the higher rank q -Bannai–Itoalgebra, which is canonically isomorphic to the higher rank Askey–Wilson algebra. In this finaligher Rank Relations for the Askey–Wilson and q -Bannai–Ito Algebra 29section we will apply the methods of Section 2 to construct this higher rank extension of the q -Bannai–Ito algebra and we will argue why the Theorems 3.1 and 3.2 have natural q -Bannai–Itoanalogs.Let us start by the defining the quantum superalgebra osp q (1 |
2) as the Z -graded associativealgebra over a field K generated by elements A + , A − , K , K − and P subject to the relations KA + K − = q / A + , KA − K − = q − / A − , { A + , A − } = K − K − q / − q − / , { P, A ± } = 0 , [ P, K ] = 0 , [ P, K − ] = 0 , KK − = K − K = 1 , P = 1 . A Casimir element is given byΓ q = (cid:32) − A + A − + q − / K − q / K − q − q − (cid:33) P. Its coproduct ∆ : osp q (1 | → osp q (1 | ⊗ and counit (cid:15) : osp q (1 | → K are defined by∆( A ± ) = A ± ⊗ KP + K − ⊗ A ± , ∆ (cid:0) K ± (cid:1) = K ± ⊗ K ± , ∆( P ) = P ⊗ P, (8.1) (cid:15) ( A ± ) = 0 , (cid:15) (cid:0) K ± (cid:1) = 1 , (cid:15) ( P ) = 1 . We will also need the definition of the q -anticommutator, given by { X, Y } q = q / XY + q − / Y X.
Now let I R be the subalgebra of osp q (1 |
2) generated by A + K , A − K , K P and Γ q and similarlywrite I L for the osp q (1 | A + K − P , A − K − P , K − P and Γ q . If one defines the algebra morphism τ R : I R → osp q (1 | ⊗ I R by τ R ( A − K ) = K P ⊗ A − K,τ R ( A + K ) = (cid:0) K − P ⊗ A + K (cid:1) + q − / (cid:0) q − q − (cid:1)(cid:0) A P ⊗ A − K (cid:1) + q − / (cid:0) q / − q − / (cid:1)(cid:0) A + K − P ⊗ K P (cid:1) + q − / (cid:0) q − q − (cid:1)(cid:0) A + K − P ⊗ Γ q (cid:1) ,τ R (cid:0) K P (cid:1) = 1 ⊗ K P − (cid:0) q − q − (cid:1) ( A + K ⊗ A − K ) ,τ R (cid:0) Γ q (cid:1) = 1 ⊗ Γ q , (8.2)then I R is readily checked to be a left coideal subalgebra of osp q (1 |
2) and a left osp q (1 | τ R . Similarly, the subalgebra I L is a right coideal subalgebra of osp q (1 | osp q (1 | τ L : I L → I L ⊗ osp q (1 |
2) defined by τ L (cid:0) A − K − P (cid:1) = A − K − P ⊗ K − P,τ L (cid:0) A + K − P (cid:1) = A + K − P ⊗ K P − q / (cid:0) q − q − (cid:1) A − K − P ⊗ A P − q / (cid:0) q / − q − / (cid:1) K − P ⊗ A + K − q / (cid:0) q − q − (cid:1) Γ q ⊗ A + K,τ L (cid:0) K − P (cid:1) = K − P ⊗ − (cid:0) q − q − (cid:1) A − K − P ⊗ A + K − P,τ L (cid:0) Γ q (cid:1) = Γ q ⊗ . (8.3)In analogy to Corollary 2.1, one can easily verify that the element ∆(Γ q ) ∈ I L ⊗ I R lies in thecotensor product of I L and I R , i.e.,(1 ⊗ τ R )∆ (cid:0) Γ q (cid:1) = ( τ L ⊗ (cid:0) Γ q (cid:1) . This observation makes it possible to repeat the extension processes outlined in Section 2 forthe q -Bannai–Ito case, and in particular to state the following definition.0 H. De Clercq Definition 8.1.
The q -Bannai–Ito algebra of rank n − osp q (1 | ⊗ n ge-nerated by the elements Γ qA with A ⊆ [1; n ], which are constructed by the left, right or mixedextension processes of Definitions 2.3, 2.5 and 2.6, upon replacing Λ ∈ U q ( sl ) by Γ q ∈ osp q (1 | τ R and τ L by the morphisms (8.1), (8.2) and (8.3).It was explained in [8, Section 2.3] that the rank n − q -Bannai–Ito algebra is isomorphic tothe Askey–Wilson algebra of the same rank. The isomorphism is explicitly given byΛ A (cid:55)→ − i (cid:0) q − q − (cid:1) Γ qA , q (cid:55)→ iq / , where i ∈ K is a square root of − q -Bannai–Ito relations (cid:8) Γ q { , } , Γ q { , } (cid:9) q = Γ q { , } + (cid:0) q / + q − / (cid:1)(cid:0) Γ q { } Γ q { , , } + Γ q { } Γ q { } (cid:1) , (cid:8) Γ q { , } , Γ q { , } (cid:9) q = Γ q { , } + (cid:0) q / + q − / (cid:1)(cid:0) Γ q { } Γ q { , , } + Γ q { } Γ q { } (cid:1) , (cid:8) Γ q { , } , Γ q { , } (cid:9) q = Γ q { , } + (cid:0) q / + q − / (cid:1)(cid:0) Γ q { } Γ q { , , } + Γ q { } Γ q { } (cid:1) , as shown in [15], to higher rank. This leads to the following analog of Theorems 3.1 and 3.2. Theorem 8.1.
Let
A, B ⊆ [1; n ] be such that B ⊆ A , then Γ qA and Γ qB commute. Theorem 8.2.
Let A , A , A and A be ( potentially empty ) subsets of [1; n ] satisfying A ≺ A ≺ A ≺ A , where we have used Definition . The relation (cid:8) Γ qA , Γ qB (cid:9) q = Γ q ( A ∪ B ) \ ( A ∩ B ) + (cid:0) q / + q − / (cid:1)(cid:0) Γ qA ∩ B Γ qA ∪ B + Γ qA \ ( A ∩ B ) Γ qB \ ( A ∩ B ) (cid:1) is satisfied for A and B defined by one of the relations (3.1) – (3.3) . In this paper, we have presented several construction techniques for the higher rank Askey–Wilson algebra AW( n ), equivalent with the algorithm given in [8]. The key observation wasthe existence of a novel, right coideal comodule subalgebra of U q ( sl ). We have proven a largeclass of algebraic identities inside AW( n ), culminating in Theorems 3.1 and 3.2, by elementaryand intrinsic methods. Each of the proofs also applies, mutatis mutandis, to the higher rank q -Bannai–Ito algebra, which is isomorphic to AW( n ).At present, it is still an open question whether the relations obtained in Theorems 3.1 and 3.2define the algebras AW( n ) abstractly. Using computer algebra packages, we have obtained allsubsets A ⊆ { , . . . , n } for which the relation ( ∗ ) is satisfied, for several values of n , and eachof them turned out to be of the form (3.1), (3.2) or (3.3). Hence if supplementary relations areneeded to obtain a complete set of defining relations for the algebras AW( n ), they will mostlikely not be of the form ( ∗ ). Other related problems are the construction of similar algebrasfor more general quantum groups U q ( g ) and the behavior of the obtained algebras at q a rootof unity different from 1. We plan to look into these questions in further research.igher Rank Relations for the Askey–Wilson and q -Bannai–Ito Algebra 31 Acknowledgements
HDC is a PhD Fellow of the Research Foundation Flanders (FWO). This work was also supportedby FWO Grant EOS 30889451. The author wishes to thank the anonymous referees for theirvaluable suggestions and comments.
References [1] Abrams L., Weibel C., Cotensor products of modules,
Trans. Amer. Math. Soc. (2002), 2173–2185,arXiv:math.RA/9912211.[2] Baseilhac P., Deformed Dolan–Grady relations in quantum integrable models,
Nuclear Phys. B (2005),491–521, arXiv:hep-th/0404149.[3] Baseilhac P., An integrable structure related with tridiagonal algebras,
Nuclear Phys. B (2005), 605–619, arXiv:math-ph/0408025.[4] Baseilhac P., Martin X., Vinet L., Zhedanov A., Little and big q -Jacobi polynomials and the Askey–Wilsonalgebra, Ramanujan J. , to appear, arXiv:1806.02656.[5] Cherednik I., Double affine Hecke algebras and Macdonald’s conjectures,
Ann. of Math. (1995), 191–216.[6] Cramp´e N., Gaboriaud J., Vinet L., Zaimi M., Revisiting the Askey–Wilson algebra with the universal R -matrix of U q ( sl (2)), arXiv:1908.04806.[7] De Bie H., De Clercq H., The q -Bannai–Ito algebra and multivariate ( − q )-Racah and Bannai–Ito polyno-mials, arXiv:1902.07883.[8] De Bie H., De Clercq H., van de Vijver W., The higher rank q -deformed Bannai–Ito and Askey–Wilsonalgebra, Comm. Math. Phys. , to appear, arXiv:1805.06642.[9] De Bie H., Genest V.X., Lemay J.-M., Vinet L., A superintegrable model with reflections on S n − and thehigher rank Bannai–Ito algebra, J. Phys. A: Math. Theor. (2017), 195202, 10 pages, arXiv:1601.07642.[10] De Bie H., Genest V.X., van de Vijver W., Vinet L., A higher rank Racah algebra and the Z n Laplace–Dunkloperator,
J. Phys. A: Math. Theor. (2018), 025203, 20 pages, arXiv:1610.02638.[11] De Bie H., Genest V.X., Vinet L., The Z n Dirac–Dunkl operator and a higher rank Bannai–Ito algebra,
Adv. Math. (2016), 390–414, arXiv:1511.02177.[12] Dunkl C.F., Differential-difference operators associated to reflection groups,
Trans. Amer. Math. Soc. (1989), 167–183.[13] Frappat L., Gaboriaud J., Ragoucy E., Vinet L., The q -Higgs and Askey–Wilson algebras, Nuclear Phys. B (2019), 114632, 13 pages, arXiv:1903.04616.[14] Frappat L., Gaboriaud J., Ragoucy E., Vinet L., The dual pair ( U q ( su (1 , , o q / (2 n )), q -oscillators andAskey–Wilson algebras, arXiv:1908.04277.[15] Genest V.X., Vinet L., Zhedanov A., The quantum superalgebra osp q (1 |
2) and a q -generalization of theBannai–Ito polynomials, Comm. Math. Phys. (2016), 465–481, arXiv:1501.05602.[16] Granovskii Ya.I., Zhedanov A.S., Hidden symmetry of the Racah and Clebsch-Gordan problems for thequantum algebra sl q (2), J. Group Theory in Physics (1993), 161–171, arXiv:hep-th/9304138.[17] Granovskii Ya.I., Zhedanov A.S., Linear covariance algebra for SL q (2), J. Phys. A: Math. Gen. (1993),L357–L359.[18] Huang H.-W., Finite-dimensional irreducible modules of the universal Askey–Wilson algebra, Comm. Math.Phys. (2015), 959–984, arXiv:1210.1740.[19] Huang H.-W., An embedding of the universal Askey–Wilson algebra into U q ( sl ) ⊗ U q ( sl ) ⊗ U q ( sl ), NuclearPhys. B (2017), 401–434, arXiv:1611.02130.[20] Iliev P., Bispectral commuting difference operators for multivariable Askey–Wilson polynomials,
Trans.Amer. Math. Soc. (2011), 1577–1598, arXiv:0801.4939.[21] Ito T., Terwilliger P., Double affine Hecke algebras of rank 1 and the Z -symmetric Askey–Wilson relations, SIGMA (2010), 065, 9 pages, arXiv:1001.2764.[22] Koekoek R., Lesky P.A., Swarttouw R.F., Hypergeometric orthogonal polynomials and their q -analogues, Springer Monographs in Mathematics , Springer-Verlag, Berlin, 2010. [23] Koornwinder T.H., Zhedanov’s algebra AW(3) and the double affine Hecke algebra in the rank one case.II. The spherical subalgebra,
SIGMA (2008), 052, 17 pages, arXiv:0711.2320.[24] Koornwinder T.H., Mazzocco M., Dualities in the q -Askey scheme and degenerate DAHA, Stud. Appl. Math. (2018), 424–473, arXiv:1803.02775.[25] Nomura K., Tridiagonal pairs and the Askey-Wilson relations,
Linear Algebra Appl. (2005), 99–106.[26] Terwilliger P., Two linear transformations each tridiagonal with respect to an eigenbasis of the other,
LinearAlgebra Appl. (2001), 149–203, arXiv:math.RA/0406555.[27] Terwilliger P., Two relations that generalize the q -Serre relations and the Dolan–Grady relations,in Physics and Combinatorics 1999 (Nagoya), World Sci. Publ., River Edge, NJ, 2001, 377–398,arXiv:math.QA/0307016.[28] Terwilliger P., Leonard pairs and the q -Racah polynomials, Linear Algebra Appl. (2004), 235–276,arXiv:math.QA/0306301.[29] Terwilliger P., The universal Askey–Wilson algebra,
SIGMA (2011), 069, 24 pages, arXiv:1104.2813.[30] Terwilliger P., The q -Onsager algebra and the universal Askey–Wilson algebra, SIGMA (2018), 044,18 pages, arXiv:1801.06083.[31] Terwilliger P., Vidunas R., Leonard pairs and the Askey–Wilson relations, J. Algebra Appl. (2004), 411–426, arXiv:math.QA/0305356.[32] Vidunas R., Askey–Wilson relations and Leonard pairs, Discrete Math. (2008), 479–495,arXiv:math.QA/0511509.[33] Zhedanov A., “Hidden symmetry” of Askey–Wilson polynomials,
Theoret. and Math. Phys.89