Racah problems for the oscillator algebra, the Lie algebra sl n , and multivariate Krawtchouk polynomials
aa r X i v : . [ m a t h . R T ] S e p RACAH PROBLEMS FOR THE OSCILLATOR ALGEBRA,THE LIE ALGEBRA sl n ,AND MULTIVARIATE KRAWTCHOUK POLYNOMIALS NICOLAS CRAMP´E, WOUTER VAN DE VIJVER, AND LUC VINET
Abstract.
The oscillator Racah algebra R n ( h ) is realized by the intermedi-ate Casimir operators arising in the multifold tensor product of the oscillatoralgebra h . An embedding of the Lie algebra sl n − into R n ( h ) is presented.It relates the representation theory of the two algebras. We establish theconnection between recoupling coefficients for h and matrix elements of sl n -representations which are both expressed in terms of multivariate Krawtchoukpolynomials of Griffiths type. Contents
1. Introduction 12. The oscillator algebra h and the Racah algebra 43. Embedding of sl n − into R n ( h ) 73.1. Embedding of sl into R ( h ) 73.2. Embedding of sl into R ( h ) 83.3. Embedding of sl n − into R n ( h ) 93.4. Labelling Abelian algebras 124. Connection between recoupling coefficients for h and sl n − -representations 144.1. sl and the Krawtchouk polynomials 154.2. sl n − and the multivariate Krawtchouk polynomials 164.3. 6 j - and 9 j -symbols 194.4. Automorphisms of R n ( h ) and sl n − Introduction
This paper studies the oscillator Racah algebra R n ( h ) viewed as the centralizerof the diagonal action of the oscillator algebra h [42] in the n-fold tensor product ofits universal algebra. We shall find that it admits an embedding of sl n − . Buildingupon that result, we shall connect the facts that the multivariate Krawtchouk Date : September 30, 2019. polynomials of Griffiths arise as 3( n − j symbols of h as well as matrix elementsof the restriction to the group O( n +1) of the symmetric representations of SU( n +1).There is growing interest in Racah algebras. These are, in particular, identifiedin the framework of Racah problems where one looks at the recouplings of tensorproducts of certain Lie algebras. We shall denote by n the number of factors.The cases with n = 3 for the Lie algebra su (2) (or su (1 , U q ( sl ) and the Lie superalgebra osp (1 |
2) have first been examined. They have ledrespectively to the (universal versions of the) Racah algebra R (3) [26, 19, 21] theAskey-Wilson algebra AW (3) [27, 30] and the Bannai-Ito algebra BI (3) [20]. Inthis picture, where there is an implicit map from the abstract Racah algebra ontothe centralizer of the diagonal action of say, su (2), U q ( sl ) or osp (1 |
2) on their tripleproduct, the images of the three generators of the Racah algebra are expressed interms of the intermediate Casimir elements. The representations of these algebrasencompass the bispectral properties of the orthogonal polynomials bearing the samename that are essentially the Racah or 6j- coefficients of the corresponding algebraswhose triple tensor products are considered. In fact this is how the AW (3) wasfirst identified [49] through its realization in terms of the recurrence and q-differenceoperators of the Askey-Wilson polynomials. These Racah algebras have arisen innumerous contexts. They have appeared as symmetry algebras of superintegrablemodels [22, 11], are featuring centrally in aspects of algebraic combinatorics [45]and are related to the Leonard pairs [44]. They have been related to Double AffineHecke Algebras (DAHA) [37, 38, 43] and degenerate cases [23]. Algebras over threestrands such as the Temperley-Lieb or Brauer ones that arise in Schur-Weyl dualityhave been shown to be quotients of Racah algebras [6, 5]. Isomorphisms with certainKauffman-Skein algebras have been established [3, 4]. Howe duality could be usedto relate different presentations [17, 18, 16]. Truncated reflection algebras attachedto U q ( ˆ sl ), to loop algebra or to the Yangian of sl have also been found [1, 2, 7] tolead to AW (3) or Racah algebras. Finally, the description of the Bannai-Ito andAskey-Wilson algebras has been cast recently in the framework of the universal R -matrix [8, 9]. This offers sufficient cause already to warrant the exploration ofthe Racah algebra associated to the oscillator algebra.The study of Racah algebras as centralizers of n -fold tensor products with n larger than 3 has been pursued [12, 13, 10, 40]. The recoupling coefficients inthese instances are orthogonal polynomials in many variables. In the case of thegeneralized Racah algebra for example, bases for representations are obtained bydiagonalizing the generators of different maximal Abelian subalgebras [12] and theconnection coefficients between two such bases are given in terms of multivariateRacah polynomials of the Tratnik type [46, 25]. Given that the Racah polynomialssit at the top of the finite part of the q=1 Askey scheme [34, 35], these Tratnikpolynomials provide, through specializations and limits, multivariable extensions ofall the finite families of orthogonal polynomials in parallel with what occurs in theunivariate situation.We shall here examine the oscillator Racah algebra R n ( h ) for arbitrary n therebyexploring the structure that encodes the properties of the 3 nj -symbols of the oscil-lator algebra h . As will be seen, these are given in terms of multivariate Krawtchoukpolynomials. Historically, the 3 j - and 6 j - coefficients were obtained in [47, 33] andfound to be given both in terms of univariate Krawtchouk polynomials. Lookingat the 9 j - symbols of h , Zhedanov made the observation [50] that these involve ACAH PROBLEMS FOR THE OSCILLATOR ALGEBRA 3 polynomials in two variables orthogonal with respect to the trinomial distributionand depending on one more parameter than the Tratnik ones. We shall indicatebelow how this generalizes.Regarding multivariate Krawtchouk polynomials, it is worth recalling and clar-ifying the following points. Polynomials in n variables that are orthogonal withrespect to the multinomial distribution were introduced by Griffiths in 1971 usinga generating function [28]; these polynomials involve n ( n −
1) parameters. For areview see [14]. The specialization to the Krawtchouk family of the Racah poly-nomials in n variables introduced by Tratnik in 1991 and mentioned before, yieldspolynomials also orthogonal with respect to the multinomial distribution but de-pending in this case on n parameters only (in addition to the maximal degree N )[25]. These two sets are hence not the same and their relation remained unclearfor some time largely because of their intricate parametrizations. The bivariateKrawtchouk polynomials of Griffiths were rediscovered in 2008 in connection witha probabilistic model and as limits of the 9 j - symbols of su (2) [29]; they were calledRahman polynomials for a while. Slightly before, Mizukawa and Tanaka [39] hadrelated the Griffiths polynomials to character algebras and provided an explicitformula in terms of Gel’fand-Aomoto hypergeometric series.Of special relevance to the present article is the group theoretical interpretationof the multivariable Krawtchouk polynomials of Griffiths that was given by Genest,Vinet and Zhedanov in [24] where they observe that these polynomials arise in thematrix elements of the representations of the orthogonal group O( n + 1) that acton the energy eigenspaces of the isotropic ( n + 1) − dimensional harmonic oscilla-tor. In other words, they have shown that the matrix elements of the restriction toO( n +1) of the symmetric representations of SU( n +1) are expressed in terms of theKrawtchouk polynomials of Griffiths; the parameters of the polynomials are thusinterpreted as the n ( n −
1) parameters, for instance the Euler angles, that specifyrotations in ( n + 1) dimensions. This cogent picture has allowed for a completecharacterization on algebraic grounds of the Griffiths polynomials (recurrence rela-tions, difference equations, generating function etc.) using the covariance propertiesof the oscillator creation and annihilation operators under O( n + 1). Furthermorethis approach clarified the connection between the Griffiths and Tratnik classes ofKrawtchouk polynomials by making explicit that the latter is simply a special caseof the former. For example, in the bivariate case ( n = 3), while the Griffiths poly-nomials with 3 parameters correspond to a general rotation in three dimensions,the Tratnik ones with 2 parameters, arise from rotations that are only products oftwo planar rotations about perpendicular axes. Related to this group theoreticalinterpretation is the work of Iliev and Terwilliger [31, 32] (see also [41]) where theKrawtchouk polynomials appear as overlap coefficients between basis elements fortwo modules of sl n +1 ( C ) with the basis elements for the representation spaces de-fined as eigenvectors of two Cartan subalgebras related by an anti-automorphismspecified by the parameters.The embedding of sl n − into R n ( h ) that we shall construct will provide, besidesits intrinsic algebraic interest, a connection between these two manifestations of themultivariate Krawtchouk polynomials of Griffiths in matrix elements of represen-tations and in recoupling coefficients. N. CRAMP´E, W. VAN DE VIJVER, AND L. VINET
The paper is organized as follows. In section 2 we introduce the Racah algebrafor the oscillator algebra. We also exhibit some properties and find a number ofcommutation relations that are needed to prove the main theorem of this papergiven in the following section. In section 3 we show how to embed the special linearLie algebra sl n − into the Racah algebra for the oscillator algebra. We then studyAbelian subalgebras of the Racah algebra for the oscillator algebra related to Cartanalgebras of the special linear algebra. These are called labelling Abelian algebrasand will be the main tool for section 4. In this section we connect the representationtheories of sl n − and of the Racah algebra for the oscillator algebra. We show howmultivariate Krawtchouk polynomials both of Tratnik type and of Griffiths typeappear as overlap oefficients between bases of irreducible representations diagonal-ized by the labelling Abelian algebras. We focus briefly on the relation with the6 j - and 9 j -symbols. We finish this section by constructing for a number of overlapsan isomorphism of sl n − and the corresponding rotation matrix. We explain thelink of these rotation matrices with the multivariate Krawtchouk polynomials ofGriffiths type. A brief conclusion follows. Appendix A records for reference howthe overlaps between representation eigenbases associated to equivalent sl Cartangenerators are obtained in terms of univariate Krawtchouk polynomials.2.
The oscillator algebra h and the Racah algebra The oscillator algebra h is the Lie algebra generated by four elements A ± , A and a central element a with following defining relations:(1) [ A − , A + ] = a, [ A , A ± ] = ± A ± . The Casimir element Q is contained in the universal enveloping algebra U ( h ) andis given by:(2) Q := aA − A + A − . We define the elements of U ( h ) ⊗ n for 1 ≤ k ≤ nA ,k := 1 ⊗ ( k − ⊗ A ⊗ ⊗ ( n − k ) , (3) A ± ,k := 1 ⊗ ( k − ⊗ A ± ⊗ ⊗ ( n − k ) , (4) a k := 1 ⊗ ( k − ⊗ a ⊗ ⊗ ( n − k ) . (5)and for any subset non-empty K ⊂ [ n ] := { , . . . , n } (6) A ,K := X k ∈ K A ,k , A ± ,K := X k ∈ K A ± ,k , a K := X k ∈ K a k . We denote the Lie algebra (isomorphic to h ) generated by the operators A ,K , A ± ,K and a K by h K . The Casimir element of this algebra is Q K : Q K := a K A ,K − A + ,K A − ,K . (7)The operators Q K will define the algebra of interest of this article. Definition 2.1.
We define the oscillator Racah algebra R n ( h ) to be the subalgebraof U ( h ) ⊗ n generated by the elements of the set { Q K | K ⊂ [ n ] and K = ∅} . Example 2.2.
The easiest non-trivial example is given for n = 3. Then a generatoris constructed for every non-empty K ∈ [3] = { , , } . The set of generators aregiven by { Q , Q , Q , Q , Q , Q , Q } . ACAH PROBLEMS FOR THE OSCILLATOR ALGEBRA 5
Proposition 2.3.
The following relations hold in U ( h ) ⊗ n , for any non-empty K ⊂ [ n ] : [ Q K , A , [ n ] ] = 0 , [ Q K , A ± , [ n ] ] = 0 , [ Q K , a [ n ] ] = 0 . Then R n ( h ) generates a subalgebra of the centralizer of the oscillator algebra h [ n ] in U ( h ) ⊗ n . We wish to find the defining commutation relations obeyed by the generators Q K of R n ( h ). First we want to point out this lemma: Lemma 2.4.
Let { K p } p =1 ...k be a set of k disjoint subsets of [ n ] . Define K B := ∪ q ∈ B K q with B ⊂ [ k ] . Consider the following map θ : R k ( h ) → R n ( h ) : Q B Q K B . This is an injective morphism. We denote its image by R K ,...,K k k ( h ) . This algebrais isomorphic to R k ( h ) . Example 2.5.
Consider a partition of the set { , , , } . For example take K = { } , K = { , } and K = { } . Then we have the following injective morphism of R ( h ) into R ( h ): θ ( Q ) = Q K = Q , θ ( Q ) = Q K K = Q ,θ ( Q ) = Q K = Q , θ ( Q ) = Q K K = Q ,θ ( Q ) = Q K = Q , θ ( Q ) = Q K K = Q ,θ ( Q ) = Q K K K = Q . Here we introduced the shortened notation KL := K ∪ L for sets K and L . Proof.
We repeat the strategy in [12, section 4.2] and generalize to the n -fold tensorproduct space. In formulas (6) we constructed an algebra h K isomorphic to h actingon the components of tensor product whose indices are in K . Consider the algebragenerated by the union of h K p . U ( h ) ⊗ k ∼ = h h K p i p ∈ [ k ] . The isomorphism is defined on the generators by 1 ⊗ ( p − ⊗ X ⊗ ⊗ ( k − p ) → X K p where X is one of the generators A ± , A or a . Inside the algebra U ( h ) ⊗ k we find R k ( h ) generated by the operators Q B := Q i i ...i l with B ⊂ [ k ]. Their imagesunder this isomorphism are Q K B := Q K i K i ...K il . Hence the operators Q K B with B ⊂ [ k ] generate an algebra isomorphic to R k ( h ) inside R n ( h ). (cid:3) Using the strategy of this proof, we can always replace indices by sets in anyrelation given. For example, consider the following relation: Q = Q + Q + Q − Q − Q − Q . This can be found by straightforward calculation. Using Lemma 2.4 we automati-cally have for three disjoint sets K , L and M (8) Q KLM = Q KL + Q KM + Q LM − Q K − Q L − Q M . This gives us a number of linear dependencies between the generators Q A . Fromthese dependencies one can prove the following: N. CRAMP´E, W. VAN DE VIJVER, AND L. VINET
Lemma 2.6.
For any set K ⊂ [ n ] , it holds that Q K = X { i,j }⊂ K Q ij − ( | K | − X i ∈ K Q i . By Lemma 2.6 it suffices to find the commutation relations of the generating set { Q ij } .We also have the following lemma: Lemma 2.7.
If either K ⊂ L or L ⊂ K or K ∩ L = ∅ then Q K and Q L commute.Proof. By construction we know that [ Q , Q ] = 0 and [ Q , Q ] = 0. Replacingthe indices by sets by Lemma 2.4 concludes the proof. (cid:3) In particular the elements Q [ n ] and Q i are central in R n ( h ). This also meansthat [ Q ij , Q lk ] = 0 if { i, j } = { l, k } or { i, j } ∩ { l, k } = ∅ . The operators Q ij and Q lk do not commute only if they have exactly one index in common. Investigation bycomputer shows that it is not possible to write [ Q ij , Q jk ] as a linear combinationof the generators Q K .The set of commutators { [ Q ij , Q jk ] } is also not linear independent. First wehave by Lemma 2.70 = [ Q ij , Q ijk ]= [ Q ij , Q ij + Q ik + Q jk − Q i − Q j − Q k ]= [ Q ij , Q jk ] + [ Q ij , Q ik ] . (9)In the second line we used formula (8). We conclude that [ Q ij , Q jk ] = [ Q ik , Q ij ].Similarly on can show that [ Q jk , Q ij ] = [ Q ik , Q jk ] by considering [ Q jk , Q ijk ] or,equivalently, switching the indices i ↔ k . We conclude for all i , j , k in [ n ]:(10) [ Q ij , Q jk ] = [ Q jk , Q ik ] = [ Q ik , Q ij ] . Remember that the elements a i for i = 1 . . . n as defined in formula (5) are n different central operators in the algebra U ( h ) ⊗ n . A tedious computation showsthat the following linear relation also holds:(11) a i [ Q jk , Q kl ] = a j [ Q ik , Q kl ] − a k [ Q ij , Q jl ] + a l [ Q ij , Q jk ] . All double commutators are obtained from the following two expressions byswitching indices, from the properties of the commutator and from relation 10:[[ Q ij , Q jk ] , Q ij ] = a k ( a i − a j ) Q ij − a i ( a i + a j ) Q jk + a j ( a i + a j ) Q ik − ( a j + a k )( a i + a j ) Q i + ( a i + a k )( a i + a j ) Q j + ( a i − a j ) Q k , (12)[[ Q ij , Q jk ] , Q kl ] = a i a l ( Q jk − Q j − Q k ) − a j a l ( Q ik − Q i − Q k ) − a i a k ( Q jl − Q j − Q l ) + a j a k ( Q il − Q i − Q l ) . (13)It follows that the generators also satisfy the following relations for all i , j and k in [ n ]:(14) [[[ Q ij , [ Q ij , [ Q ij , Q jk ]]] = ( a i + a j ) [ Q ij , Q jk ] . Observe that relation (14) is the Dolan-Grady relation up to central elements asdefined in [15].
ACAH PROBLEMS FOR THE OSCILLATOR ALGEBRA 7
Let us generate an algebra from the set { Q K | ∅ 6 = K ⊂ [ n ] } using the Lie bracketinstead of the ordinary multiplication on U ( h ) ⊗ n . Denote it by R n ( h , []). From therelations above we conclude that R n ( h , []) is generated as a vector space by theoperators Q K and their commutators [ Q K , Q L ] over the field R ( a , . . . , a n ). Weperform some reductions. First the Q i and Q ij generate all Q K by Lemma 2.6.The commutators [ Q ij , Q jk ] can be written as a linear combination of [ Q j , Q jk ]by formula (11). By formula (9) we have [ Q j , Q jk ] = [ Q k , Q kj ] so we require that j < k . It follows then that as a vector space R n ( h , []) is generated by the followingset:(15) { Q i | i ∈ [ n ] } ∪ { Q ij | ≤ i < j ≤ n } ∪ { [ Q j , Q jk ] | < j < k ≤ n } . We will prove later on that this is a basis for R n ( h , []) as a vector space over thefield R ( a , . . . , a n ). Moreover, we will prove that the set of equalities (12), (13) and(11) together with Lemma 2.7 and Lemma 2.6 exhausts all commutation relationsof R n ( h , []). 3. Embedding of sl n − into R n ( h )In this section we study the relationship between the special linear Lie algebra sl n − and R n ( h ). The Lie algebra sl n − is generated by the following set of elements: { e kk +1 , e k +1 k , h k | ≤ k ≤ n − } . By applying Serre’s theorem on sl n − we areguaranteed of a full set of relations which we give here. To this end we introducethe Cartan matrix: A ij = i = j − j = i ±
10 if | j − i | > . Here are the Chevalley-Serre relations:[ h i , h j ] = 0 , (16) [ e ii +1 , e j +1 j ] = δ ij h i , (17) [ h i , e jj +1 ] = A ij e jj +1 , [ h i , e j +1 j ] = − A ij e j +1 j , (18) ad( e ii +1 ) − A ij ( e jj +1 ) = 0 , if i = j (19) ad( e i +1 i ) − A ij ( e j +1 j ) = 0 . if i = j (20)The operator ad is the adjoint action: ad( x )( y ) := [ x, y ]. The set { h k | ≤ k ≤ n − } generates the Cartan algebra of sl n − . When we consider the sl case, thereare three generators { e , e , h } with following relations:(21) [ e , e ] = h , [ h , e ] = 2 e , [ h , e ] = − e . From the Chevalley-Serre relations we see that every triple { e ii +1 , e i +1 i , h i } gener-ates a copy of sl .3.1. Embedding of sl into R ( h ) . Consider the algebra R ( h ) and consider theadjoint action of Q on R ( h ). We want to find its eigenspaces. The eigenspacewith eigenvalue 0 is a five-dimensional space generated by the central elements { Q , Q , Q , Q } and the operator Q . The eigenvectors with nonzero eigenvalue N. CRAMP´E, W. VAN DE VIJVER, AND L. VINET are e := λ ([ Q , [ Q , Q ]] + ( a + a )[ Q , Q ]) ,e := λ ([ Q , [ Q , Q ]] − ( a + a )[ Q , Q ]) . (22)We introduced the number λ = 1 p a a a ( a + a ) ( a + a + a ) . One checks easily using relation (14) that:[ Q , e ] = ( a + a ) e , [ Q , e ] = − ( a + a ) e . (23)We also define the following operator(24) h := 2 Q a + a − Q a + a + a − Q a − Q a + Q a . We have the following proposition:
Proposition 3.1.
The operators e , e and h satisfy the commutation relationsof sl . [ e , e ] = h , [ h , e ] = 2 e , [ h , e ] = − e Proof.
One checks through straightforward calculation. (cid:3)
By Proposition 3.1 we have a map of sl into R ( h ). Because sl is simple, thekernel of this map is either trivial of equal to the whole algeba sl . Clearly, e , e and h are different from 0 so the kernel must be trivial. This map must thereforebe injective and we have indeed an embedding of sl into R ( h ).3.2. Embedding of sl into R ( h ) . Consider the algebra R ( h ). We want to findthe common eigenvectors of Q and Q . By Lemma 2.7 the operators e , e and h commute with Q and are therefore eigenvectors of both Q and Q .We can find another set of eigenvectors using Lemma 2.4. Consider the operators e , e and h expressed in the operators Q , Q and a and replace the indicesas follows by Lemma 2.4: 1 → { , } , 2 → →
4. We find the followingoperators: e := λ ([ Q , [ Q , Q ]] + ( a + a + a )[ Q , Q ]) ,e := λ ([ Q , [ Q , Q ]] − ( a + a + a )[ Q , Q ]) ,h := 2 Q a + a + a − Q a + a + a + a − Q a + a − Q a + Q a . The number λ is given by λ = 1 p a + a ) a a ( a + a + a ) ( a + a + a + a ) . By Lemma 2.7 the operators e , e and h commute with Q . They are alsoeigenvectors of Q by formula 23:[ Q , e ] = ( a + a + a ) e , [ Q , e ] = − ( a + a + a ) e . (25)The operators e , e and h satisfy the sl relations:[ e , e ] = h , [ h , e ] = 2 e , [ h , e ] = − e . ACAH PROBLEMS FOR THE OSCILLATOR ALGEBRA 9
We have the following claim:
Proposition 3.2.
The operators { e , e , e , e , h , h } satisfy the commutationrelations of sl .Proof. We already know that both { e , e , h } and { e , e , h } satisfy the sl relations. It is also easy to show by Lemma 2.7 that[ h , h ] = 0 . By straightforward calculation [ e , e ] = 0 , [ e , e ] = 0 . By now we verified relations (16) and (17). By explicit calculation using the defi-nition of h and h and formula (23) and (25):[ h , e ] = (cid:20) − Q a + a + a , e (cid:21) = − e , [ h , e ] = (cid:20) − Q a + a + a , e (cid:21) = e , [ h , e ] = (cid:20) − Q a + a , e (cid:21) = − e , [ h , e ] = (cid:20) − Q a + a , e (cid:21) = e . Relation (18) is also satisfied. We only need check (19) and (20):[ e , [ e , e ]] = 0 , [ e , [ e , e ]] = 0 , [ e , [ e , e ]] = 0 , [ e , [ e , e ]] = 0 . This is done by straightforward computation. By Serre’s Theorem these are acomplete set of defining relations and this concludes the proof. (cid:3)
By Proposition 3.2 we have a map of sl into R ( h ). Because sl is simple, thekernel of this map is either trivial of equal to the whole algeba sl . The generatorswe used in Proposition 3.2 are different from 0. This map must therefore be injectiveand we have indeed an embedding of sl into R ( h ).3.3. Embedding of sl n − into R n ( h ) . As in the previous two sections we con-struct common eigenvectors for the adjoint action of the Abelian subalgebra Y = { Q [ k ] | ≤ k ≤ n − } . Observe that by formula (6) we have a B = P i ∈ B a i . Toconstruct the eigenvectors we use Lemma 2.4 in the following way: Take e , e and h given by formulas (22) and (24) and replace 1 by [ k ], 2 by k + 1 and 3 by k + 2. We obtain the following elements: e kk +1 := λ k ([ Q [ k +1] , [ Q [ k +1] , Q k +1 k +2 ]] + a [ k +1] [ Q [ k +1] , Q k +1 k +2 ]) ,e k +1 k := λ k ([ Q [ k +1] , [ Q [ k +1] , Q k +1 k +2 ]] − a [ k +1] [ Q [ k +1] , Q k +1 k +2 ]) ,h k := 2 Q [ k +1] a [ k +1] − Q [ k +2] a [ k +2] − Q [ k ] a [ k ] − Q k +1 a k +1 + Q k +2 a k +2 . We introduced the element λ k = 1 q a [ k ] a k +1 a k +2 a k +1] a [ k +2] . We check that these operators are indeed eigenvectors of the Abelian algebra Y .By Lemma 2.7 the operator Q [ l ] commutes with e kk +1 and e k +1 k if l = k + 1 andwith h k for all k . If l = k + 1 we have[ Q [ k +1] ,e kk +1 ]= λ k ([ Q [ k +1] , [ Q [ k +1] , [ Q [ k +1] , Q k +1 k +2 ]]] + a [ k +1] [ Q [ k +1] , [ Q [ k +1] , Q k +1 k +2 ]])= λ k ( a k ] [ Q [ k +1] , Q k +1 k +2 ] + a [ k +1] [ Q [ k +1] , [ Q [ k +1] , Q k +1 k +2 ]])= a [ k ] e kk +1 . We used formula (14). Similarly one can show that [ Q [ k +1] , e k +1 k ] = − a [ k ] e k +1 k .We are now ready to prove the following theorem. Theorem 3.3.
The set of operators { e ii +1 , e ii +1 , h i | i ∈ [ n − } ⊂ R n ( h ) generatean algebra isomorphic to sl n − for the Lie bracket.Proof. We prove this statement by induction. The cases n = 2 and n = 3 have al-ready be obtained in Propositions 3.1 and 3.2. Now assume that { e ii +1 , e ii +1 , h i | i ∈ [ n − } generates sl n − . To go to sl n − we add three new operators: e n − n − , e n − n − and h n − . We introduce a morphism using Lemma 2.4: σ : R n − ( h ) → R n ( h )by mapping the indices 1 → { , } and i → i + 1 for every i >
1. One should noticethat σ ( e k − k ) = e kk +1 , σ ( e kk − ) = e k +1 k and σ ( h k − ) = h k . It maps sl n − as asubalgebra of R n − ( h ) into R n ( h ). We will use this map a few times.We check the relations of sl n using Serre’s Theorem. By Lemma 2.4 and Propo-sition 3.1 we know that e kk +1 , e k +1 k and h k generate a sl algebra for every k :[ h k , e kk +1 ] = 2 e kk +1 , [ h k , e k +1 k ] = − e k +1 k , [ e kk +1 , e k +1 k ] = h k . We also have by Lemma 2.7: [ h k , h l ] = 0 . Relation (16) is satisfied so the set { h k | k = 1 . . . n − } plays the role Cartanalgebra of sl n − . The following relation we need to check is [ e kk +1 , e l +1 l ] = 0 if l = k . If l / ∈ { k − , k, k + 1 } this is true by Lemma 2.7. Otherwise set l = k + 1.Then we can use the map σ :[ e kk +1 , e k +2 k +1 ] = σ ([ e k − k , e k +1 k ]) = σ (0) = 0 . Here we used the relations of the algebra sl n − . The proof for l = k − h k , e jj +1 ] ACAH PROBLEMS FOR THE OSCILLATOR ALGEBRA 11 and [ h k , e j +1 j ]:[ h k , e jj +1 ] = (cid:20) Q [ k +1] a [ k +1] − Q [ k +2] a [ k +2] − Q [ k ] a [ k ] , e jj +1 (cid:21) = 2 δ k,j e jj +1 − δ k +2 ,j +1 e jj +1 − δ k,j +1 e jj +1 , [ h k , e j +1 j ] = (cid:20) Q [ k +1] a [ k +1] − Q [ k +2] a [ k +2] − Q [ k ] a [ k ] , e j +1 j (cid:21) = − δ k,j e j +1 j + δ k +2 ,j +1 e j +1 j + δ k,j +1 e j +1 j . If k / ∈ { j − , j, j + 1 } then [ h k , e jj +1 ] = 0 and [ h k , e j +1 j ] = 0. Otherwise, we have[ h j − , e jj +1 ] = − e jj +1 , [ h j +1 , e jj +1 ] = − e jj +1 , [ h j − , e j +1 j ] = e j +1 j , [ h j +1 , e j +1 j ] = e j +1 j . We have verified relation (18). Finally we need to check the relations (19) and (20).Specifically, we need to check the following:[ e jj +1 , e kk +1 ] = 0 , if k / ∈ { j − , j + 1 } , (26) [ e j +1 j , e k +1 k ] = 0 , if k / ∈ { j − , j + 1 } , (27) [ e jj +1 , [ e jj +1 , e kk +1 ]] = 0 , if k ∈ { j − , j + 1 } , (28) [ e j +1 j , [ e j +1 j , e k +1 k ]] = 0 , if k ∈ { j − , j + 1 } . (29)The relations (26) and (27) follow by Lemma 2.7. The relations (28 )and (29) canbe proven as follows.[ e jj +1 , [ e jj +1 , e kk +1 ]] = σ ([ e j − j , [ e j − j , e k − k ]]) = σ (0) = 0 , [ e j +1 j , [ e j +1 j , e k +1 k ]] = σ ([ e jj − , [ e jj − , e kk − ]]) = σ (0) = 0 . Here we used the sl n − relations. We have shown that the algebra generated by { e ii +1 , e ii +1 , h i | i ∈ [ n − } is homomorphic to sl n − . To show that it is in factisomorphic we need to prove that the map from sl n − into R n ( h ) is injective. Thisis seen by the same argument as before based on the simplicity of sl n − and thuswe have an embedding (cid:3) We state the following corollary.
Corollary 3.4.
As Lie algebras over the field K := R ( a , . . . , a n ) we have thefollowing isomorphism: R n ( h , []) ∼ = sl n − ⊕ K n +1 . Proof.
The set (15) generates R n ( h , []) as a vector space over K := R ( a , . . . , a n )but we do not know yet if this is a basis. We have n + 1 central elements, n ( n − elements of the form Q ij and ( n − n − elements of the form [ Q j , Q jk ]. We havetherefore(30) dim( R n ( h , [])) ≤ n + 1 + n ( n − n − n − n − n + 1 . By Theorem 3.3 we know there exists a subalgebra of R n ( h , []) isomorphic with sl n − . Denote this Lie algebra by s . The dimension of s equals ( n − −
1. We add to this algebra s the vector space generated by the central elements Z := h{ Q i | i ∈ [ n ] } ∪ { Q [ n ] }i . First notice that s ∩ Z = { } . The algebra s is simple so it does not contain centralelements. Second we see that { Q i | i ∈ [ n ] } ∪ { Q [ n ] } is a basis for Z . If this was notthe case, we could find a linear combination of the central elements equal to 0. Weshow that this is not possible. The Q i are linearly independent by construction.Assume we have the linear combination: Q [ n ] = n X i =1 λ i Q i . Act with µ ∗ ⊗ N n − i =1 Q . We find0 = [ Q , Q [ n +1] ] = " Q , λ Q + n X i =2 λ i Q i +1 = λ [ Q , Q ] . The element [ Q , Q ] does not equal 0 so λ = 0. Similarly one can show that all λ i = 0 concluding that { Q i | i ∈ [ n ] } ∪ { Q [ n ] } is indeed a basis for Z . The dimensionof Z equals n + 1. We have s ⊕ Z ⊂ R n ( h , []) . From this follows that n − n + 1 = dim( s ⊕ Z ) ≤ dim( R n ( h , [])) . We already have an upper bound by inequality (30). Therefore it follows thatdim( R n ( h , [])) = n − n + 1 and hence R n ( h , []) = s ⊕ Z ∼ = sl n − ⊕ K n +1 . This concludes the proof. (cid:3)
Theorem 3.3 and Corollary 3.4 both follow from the set of equations calculatedin section 2. As a consequence of the isomorphism found in Corollary 3.4 we give analternative definition of R n ( h , []). It is the algebra defined over R ( a , . . . , a n ) withgenerators Q i and Q ij satisfying the set of equalities (11), (12) and (13), togetherwith Lemma 2.7 and Lemma 2.6.3.4. Labelling Abelian algebras.
In the proof of Theorem 3.3 the elements ofthe Cartan algebra of sl n − are linear combinations of the central elements and { Q [ k ] | ≤ k ≤ n − } . It is possible to construct different Cartan algebras if onestarts from a different set of generators. Definition 3.5.
A set of non-empty subsets of [ n ] A is a maximal non-intersecting/nested set if it satisfies the following properties: • Every pair of sets in A is either disjoint or one is included in the other. • Maximality: It is not possible to add another subset of [ n ] to A withoutcontradicting the first property.By Lemma 2.7 the corresponding generators Q A with A ∈ A will be a set ofcommuting operators. To each maximally non-intersecting/nested set A we canassociate a graph T A . Let every vertex of the graph represent a set in A . There isan edge between to vertices A and B if either set is included in the other but thereis no set C ∈ A such that A ⊂ C ⊂ B or B ⊂ C ⊂ A . ACAH PROBLEMS FOR THE OSCILLATOR ALGEBRA 13
Lemma 3.6.
The graph T A related to the maximal non-intersecting/nested set A is a perfect binary tree.Proof. The set [ n ] is always included in A by maximality. The set [ n ] includesevery set in A and will be the root of the tree. The sets with one element are alsoincluded in A by maximality. They are either completely included or disjoint withevery set in A . These sets do not have any proper subsets so they cannot have anychildren and are therefore leaves of the tree. Every other vertex has two children.Assume this is not the case. Let A have exactly one child B . Then we can add A \ B to A contradicting the maximality of A . Assume A has more than two children.Let B , B and B three sets that are children of A . We can add the set B ∪ B to A contradicting the maximality of A . This concludes the proof. (cid:3) We have n indices so we have n leaves. A perfect binary tree with n leaves has n − n − Definition 3.7.
Let A be a maximally non-intersecting/nested set of [ n ]. Thenwe define the labelling Abelian algebra associated to A to be: Y A = { Q A | A ∈ A and 1 < | A | < n } . We exclude the elements Q i and Q [ n ] as they are central. For each maximallynon-intersecting/nested set A we have constructed a tree T A and an algebra Y A .Often we will represent the algebra Y A by the tree T A . These trees are similar tothe coupling trees introduced in [48]. Example 3.8.
Consider R ( h ). We have three indices 1, 2 and 3. This gives usthree possible trees and therefore three different labelling Abelian algebras:112 1232 3 Y A = { Q } Y A = { Q }
113 1233 2 Y A = { Q } Example 3.9.
Consider R ( h ). We have four indices 1, 2, 3 and 4 so we have treeswith four leaves. We consider two examples:1121232 31234 4 Y A = { Q , Q }
112 342 31234 4 Y A = { Q , Q } We are now in a position to explain the relation between these labelling Abelianalgebras and the Cartan algebra of sl n − inside R n ( h ). Assume we have fixed alabelling Abelian algebra Y A . For every set A ∈ A that is not the root or a leaf,we will construct three generators e A , f A and h A . To do this consider the tree T A .Focus on the subtree consisting of the vertex related to the set A : The children of A which we will call K and L , the parent B of A and the child of B differing from A denoted by M . We have two possibilities:KA BL M M ABK LWe define e A and f A as follows: e A := λ A ([ Q A , [ Q A , Q LM ]] + a A [ Q A , Q LM ]) ,f A := λ A ([ Q A , [ Q A , Q LM ]] − a A [ Q A , Q LM ]) . (31)with λ A = 1 p a K a L a M a A a B . Observe that it is possible to replace Q LM by Q KM in the definitions of e A and f A leading to different but equally correct expressions for e A and f A . To avoidthis ambiguity we always choose L to be the right child of A and M the childof B different from A . This, however, fixes an ordering of the leaves. Otherwiseit is not possible to speak of a left and right child. Together with the Cartanelements h A = [ e A , f A ] the set { e A , f A , h A } generates the Lie algebra sl n − . Thiscan be proven by repeating the arguments in Theorem 3.3. The maximally non-intersecting/nested set used in Theorem 3.3 is { [ k ] | < k < n } . To find the relationbetween the Cartan algebra { h A | A ∈ A and 1 < | A | < n } and the labelling Abelianalgebra Y A we consider explicitly h A : h A = 2 Q A a A − Q B a B − Q K a K − Q L a L + Q M a M . Every Cartan element can be written as a linear combination of elements of thelabelling Abelian algebra Y A and central elements Q i and Q [ n ] .In the next section we will study the representation theory behind R n ( h ) andits relation to the Lie algebra sl n − . The relation between Cartan algebras andlabelling Abelian algebras will play an important role there.4. Connection between recoupling coefficients for h and sl n − -representations Assume we have a finite dimensional irreducible representation V of R n ( h ). Be-cause of corollary 3.4 this is also an irreducible representation for sl n − . In thissection we study bases diagonalized by different labelling Abelian algebas or equiv-alently Cartan algebras of sl n − and their connection coefficients. We first studythe rank one case. ACAH PROBLEMS FOR THE OSCILLATOR ALGEBRA 15 sl and the Krawtchouk polynomials. Consider the following two Abelianalgebras of R ( h ): 112 1232 3 Y A = { Q } Y A = { Q } Let { ψ k } be an eigenbasis of Q and { φ s } be an eigenbasis of Q . The indices k and s run form 0 to N with dim( V ) = N + 1. We are interested in the overlapcoefficients B ks between these bases:(32) φ s = N X k =0 B sk ψ k . To study these overlap coefficients we construct the Lie algebra sl inside R ( h ).Corresponding to Y A , we have the following generators. See also formula (22): e := λ ([ Q , [ Q , Q ]] + a [ Q , Q ]) ,f := λ ([ Q , [ Q , Q ]] − a [ Q , Q ]) ,h := 2 Q a − Q a − Q a − Q a + Q a with λ = p a a a a a − . The sl Lie algebra related to Y A is given byreplacing indices (1 → → →
1) in e , f and h given above. We denote theseelements by ˜ e , ˜ f and ˜ h . The elements h and ˜ h diagonalize the bases { ψ k } and { φ s } respectively. hψ k = µ k ψ k , ˜ hφ s = ν s φ s . The map ˜ . is an automorphism of sl so we can write ˜ h as a linear combination of h , e and f . ˜ h = R h h + R e e + R f f with R h = − a a − a a + a + a a a a ,R e = R f = 2 √ a a a a a a . Observe that R e R f + R h = 1. It is a classical result that the overlap coefficients be-tween two bases related by an inner automorphism of sl are univariate Krawtchoukpolynomials [24, 36]. We have therefore relegated the calculations to Appendix A.This is the result: B sk ∼ K k (cid:18) ν s + N − R h , N (cid:19) . The Krawtchouk polynomials depend on 2 values: R h and N . The number N isequal by definition to dim( V ) −
1. The dimension of an irreducible representation of sl can be found by considering its Casimir: C := h ef + f e = 12 (cid:18) Q a + Q a + Q a − Q a (cid:19) (cid:18) Q a + Q a + Q a − Q a − (cid:19) . (33)If we act with the Casimir on ψ k we find: Cψ k = N + 2 N ψ k . From the action of the Casimir C of sl or equivalently the central elements Q , Q , Q and Q we are able to discern the dimension of the representation andhence the number N . For the remainder of the article we will write the dependenceon R h and C explicitly: B sk ( R h , C ).4.2. sl n − and the multivariate Krawtchouk polynomials. Let Y A and Y A be two labelling Abelian algebras of R n ( h ). Additionally we demand that A and A differ by only one element: A \A = { G } , A \A = { G } . Let { ψ ~k } be diagonalized by the labelling Abelian algebra Y A and { ψ ~s } be diago-nalized by the labelling Abelian algebra Y A . Let A = { A i | i = 1 . . . n − } , thenwe have Q A i ψ ~k = µ A i k i ψ ~k . For a specific index j it must be that A j = G . We want to find the overlapcoefficients between the bases { ψ ~k } and { φ ~s } . φ ~s = X ~k B ~s~k ψ ~k . The basis vector φ ~s is a common eigenvector of the operators Q A i , i = j witheigenvalues µ A i s i . The basis { ψ ~k } also consists of eigenvectors of the operators Q A i , i = j . The vector φ ~s must therefore be written as a linear combination of commoneigenvectors of Q A i , i = j with the same eigenvalues µ A i s i , i = j . It follows that B ~s~k = 0 if k i = s i for some i = j . The overlap coefficient can be written as B ~s~k = B s j k j Y i = j δ s i k i where δ s i k i is the Kronecker delta. To find B s j k j consider the common eigenspace T = { v ∈ V | Q A i v = µ A i s i v for all i = j } . Both Q G and Q G commute witheach Q A i so they preserve the common eigenspace T . In fact Q G and Q G lie in an algebra isomorphic to R ( h ) that preserves T . Let K = G \ G , L = G ∩ G and M = G \ G . By Lemma 2.4 the algebra R K,L,M ( h ) generated by { Q K , Q L , Q M , Q KL , Q LM , Q KLM } is isomorphic to R ( h ). It preserves T as eachgenerator commutes with Q A i with i = j . In fact the sets K , L , M and K ∪ L ∪ M are in A ∩ A . We conclude that T is a representation of R ( h ) with one basis { ψ ~k } ∩ T diagonalized by Q G and the other { φ ~s } ∩ T by Q G . We are basically inthe situation discussed in the previous paragraph and represented by the followingtwo trees: ACAH PROBLEMS FOR THE OSCILLATOR ALGEBRA 17 KG KLML M K G KLML M
This means that the overlap coefficients are given by B s j k j = B s j k j ( R K,L,Mh , C
K,L,M ) . The elements R K,L,Mh and C K,L,M are obtained by using the isomorphism between R ( h ) and R K,L,M ( h ) obtained by replacing the indices 1, 2 and 3 by K , L and M : R K,L,Mh = − a K a L − a K a M + a L + a L a M a KL a LM ,C K,L,M = 12 (cid:18) Q K a K + Q L a L + Q M a M − Q KLM a KLM (cid:19) (cid:18) Q K a K + Q L a L + Q M a M − Q KLM a KLM − (cid:19) . We have shown that the overlap coefficients between two bases diagonalized bylabelling Abelian algebras differing by one generators are Krawtchouk polynomi-als. Let us remove the condition that the labelling Abelian algebras need to differby one generator. Then the strategy to find the overlap coefficients is to find aseries of intermediate bases in such a way that each intermediate basis differs byone generator with the next intermediate basis. For example take the labellingAbelian algebras h Q , Q i and h Q , Q i in R ( h ). Then we can find a series ofintermediate bases: h Q , Q i → h Q , Q i → h Q , Q i → h Q , Q i . Each step gives us Krawtchouk polynomials as overlaps. More specifically let { ψ k k } , { ψ k l } , { ψ s l } and { φ s s } be the bases diagonalized by each step inthis chain. Then the overlap coefficients become: φ s s = X l B s l ( R , , h , C , , ) ψ s l = X l k B s l ( R , , h , C , , ) B s k ( R , , h , C , , ) ψ k l = X l k k B s l ( R , , h , C , , ) B s k ( R , , h , C , , ) B l k ( R , , h , C , , ) ψ k k . Here we conclude that the overlap coefficients are(34) B ~s~k = X l B s l ( R , , h , C , , ) B s k ( R , , h , C , , ) B l k ( R , , h , C , , ) . This gives us a method to calculate connection coefficients between any pair ofbases diagonalized by labelling Abelian algebras. We do need to check if there isalways a path between two labelling Abelian algebras. To this end we introduce therecoupling graph of R n ( h ). Let every labelling Abelian algebra be represented by avertex. Two vertices are connected by an edge if they only differ by one generator. Proposition 4.1.
The recoupling graph of R n ( h ) is connected and its diameter isbounded by ( n − n − . Proof.
We use an argument applied on the binary trees related to labelling Abelianalgebras. On these trees we have two operations: • A twist: interchanging two children (and their descendant trees) of a vertex.This does not change the labelling Abelian algebra so on the connectiongraph we stay on the same vertex.
A B → B A • A swap: Moving a vertex’ child to the other edge. This changes the labellingAbelian algebra by one generator. On the connection graph we move alongan edge to a new vertex. → Because twisting does not change the labelling Abelian algebra, you stay on thesame vertex of the connection graph. Every swap on the other hand is related to astep on the connection graph. The proof of connectedness in ( n − n − steps usesinduction:Let n = 3 and take any two trees with three leaves. It is easy to see that it takesa single swap combined with a number of twists to get from one tree to the other.Assume we have proved it for n −
1. We take two trees with n leaves. We callthese trees the initial tree and final tree. In the initial tree there is at least onevertex whose children are leaves. Assume these leaves are labelled a and b . Removethose leaves and give the parent vertex the label a . We now have a tree with n − b . Remove this label b also from the finaltree together with its parent. The final tree now also has n − ( n − n − swaps and a number of twists to change the initialtree to the final tree. Add the leaf with label b and its parent again to the finaltree where it was removed. In the initial tree we add two leaves to the leaf withlabel a . We remove this label a and add label a and b to the leaves. Only leaf b isin the wrong place. There are at most n − b and were itneeds to be in the final tree. It requires n-2 swaps and a number of twists to moveleaf b into the right position. The total swaps used to change the initial tree intothe final tree is ( n − n − n − n − n − . (cid:3) Remark 4.2.
This proof also works for the higher rank Racah algebra for su (1 , asin [12] effectively generalizing the connection graph and the proof of connectednessof the connection graph in [12] . We want to conclude this section with a special pair of labelling Abelian algebrasof Y = { Q [ k ] | ≤ k ≤ n − } and Y n − = { Q [2 ...k ] | ≤ k ≤ n } . Observe that Y n − can be obtained by from Y by cyclicly permuting the indices i → i + 1 and n →
1. We can find a path between these two algebras as follows. Define Y l := { Q [ k ] | l + 1 ≤ k ≤ n − } ∪ { Q [2 ...k ] | ≤ k ≤ l + 1 } . ACAH PROBLEMS FOR THE OSCILLATOR ALGEBRA 19
These are labelling Abelian algebra with Y l − \Y l = Q [ l ] and Y l \Y l − = Q [2 ...l +1] .If we determine the overlap associated to each step we find the following connectioncoefficients: B ~s~k = n − Y l =2 B s l k l ( R , [2 ..l ] ,l +1 h , C , [2 ..l ] ,l +1 ) . These are the multivariate Krawtchouk polynomials of Tratnik type. They dependon n − R , [2 ..l ] ,l +1 h . They are constructed in the same way as the mul-tivariate Racah polynomials were constructed as connections coefficients betweenlabelling Abelian algebras for the Racah algebra in [12]. The number of parametersfollows from the number of steps we needed to move through the connection graph.By proposition 4.1 we know that we can find paths up to ( n − n − steps. Furtherin the paper we will show how to get the multivariate Krawtchouk polynomials ofGriffiths type depending on ( n − n − parameters.4.3. 6 j - and j -symbols. The 6 j -, 9 j - and in general the 3 nj -symbols for theoscillator algebra h can be cast into the framework presented in this article as theyare specific overlaps between recoupled bases. Consider the 6 j -symbols. Given thealgebra h ⊕ h ⊕ h , the 6 j -symbols or Racah coefficients are the connection coefficientsfor coupling the first two oscillator algebras and the last two:( h ⊕ h ) ⊕ h → h ⊕ ( h ⊕ h ) . In our framework this is equivalent with finding the overlap between the basesdiagonalized by Q and Q . The 6 j -symbols are therefore up to a normalizationequal to B sk ( R h , C ). One could ask where the 6 j ’s of the 6 j -symbols are. Theindices s and k are related to the eigenvalues of Q and Q so they are relatedto j and j . The number R h is independent of the representation used. Theremaining four j ’s are hidden in the Casimir C . According to formula (2) theCasimir depends on Q , Q , Q and Q giving the remaining j , j , j and j .A similar analysis can be given for the 9 j -symbols. The 9 j -symbols were al-ready identified as bivariate Krawtchouk polynomials in [50]. They are obtainedby considering the algebra h ⊕ h ⊕ h ⊕ h and the following two bases: the first basisobtained by coupling first factor with the second and the third with the fourth andthe second basis defined by coupling the first factor with the third and the secondwith the fourth. i.e. (1 ⊕ ⊕ (3 ⊕ → (1 ⊕ ⊕ (2 ⊕ . In our framework this is equivalent with finding the overlap coefficients betweenthe bases diagonalized by h Q , Q i and h Q , Q i . We already calculated theconnection coefficients in formula (34). We repeat the solution here: B ~s~k = X l B s l ( R , , h , C , , ) B s k ( R , , h , C , , ) B l k ( R , , h , C , , ) . These numbers are multivariate Krawtchouk polynomials depending on three pa-rameter R , , h , R , , h and R , , h . These numbers B ~s~k are up to normalizationequal to the 9 j -symbols. The 9 j ’s are found in the following way: ~s and ~k arerelated to the eigenvalues of { Q , Q } and { Q , Q } respectively. The othersare found by considering the three Casimir elements appearing in the formula. We can add the following generators: { Q , Q , Q , Q , Q , Q } . This gives a total of 10 numbers. This is one too many. The summation runs over l which appears not only as an index but also in C , , . This number which isrelated to the eigenvalues of Q can be considered as being summed away. Thisleaves us with 9 generators related to the 9 j ’s in the 9 j -symbols. A similar analysiscan be done for any 3 nj -symbol.4.4. Automorphisms of R n ( h ) and sl n − . For each labelling Abelian algebra of R n ( h ) we are able to construct a set of operators that generate sl n − . By Corollary3.4 these sets of operators must generate the same algebra. This leads to automor-phisms of sl n − . In this section we will give a few examples. Additionally we willgive the group element of Lie group SO n − corresponding to each automorphism. Example 4.3.
Take the algebra R ( h ) and construct sl from the labelling Abelianalgebra { Q } : As per formula (31): e := λ ([ Q , [ Q , Q ]] + a [ Q , Q ]) ,f := λ ([ Q , [ Q , Q ]] − a [ Q , Q ]) ,h := [ e, f ] . Consider the permutation (1 ↔ →
212 1231 3Keeping in mind that [ Q , Q ] = − [ Q , Q ] by formula (9), the images of thegenerators under this permutation are ˜ h = h , ˜ e = − e and ˜ f = − f . The automor-phism of SL is constructed as follows. Let(35) U (cid:18) h fe − h (cid:19) U − = ˜ h ˜ f ˜ e − ˜ h ! . We want to solve for U with U ∈ SL . Conjugation by U is the related automor-phism of SL . In this case U (12) = (cid:18) − (cid:19) . Example 4.4.
Next we construct sl from the labelling Abelian algebra { Q } .We are constructing an isomorphism corresponding to these two trees:112 1232 3 → ACAH PROBLEMS FOR THE OSCILLATOR ALGEBRA 21
This is a swap as in Proposition 4.1. We obtain the second set of generators bypermuting the permutation (1 → → → h , ˜ e and ˜ f are linear combinations of the original generators h , e and f :˜ h = 2 √ a a a a a a ( e + f ) + a a + a a + a − a a a a h, ˜ e = a a a a e − a a a a f − √ a a a a a a h, ˜ f = − a a a a e + a a a a f − √ a a a a a a h. This set of equalities gives an automorphism of sl . We find the correspondinggroup element of SO by solving the following matrix equation:(36) U (cid:18) h fe − h (cid:19) U − = ˜ h ˜ f ˜ e − ˜ h ! . Solving for U gives U (123) = q a a a a − q a a a a q a a a a q a a a a . A straightforward calculation shows that U (123) is indeed an orthogonal matrix. Itrepresents a rotation with angle θ , , defined bycos( θ , , ) = r a a a a . Example 4.5.
Consider in the algebra R ( h ) the Lie algebra sl constructed fromthe labelling Abelian algebra { Q , Q } . e := λ ([ Q , [ Q , Q ]] + a [ Q , Q ]) ,e := λ ([ Q , [ Q , Q ]] − a [ Q , Q ]) ,h := [ e , e ] ,e := λ ([ Q , [ Q , Q ]] + a [ Q , Q ]) ,e := λ ([ Q , [ Q , Q ]] − a [ Q , Q ]) ,h := [ e , e ] ,e := [ e , e ] ,e := [ e , e ] . When performing the permutation (1 → → →
1) we find new generators. Thesegenerators are eigenvectors of the labelling Abelian algebra { Q , Q } . We willnot explicitly express the new generators as linear combinations but we will give U ∈ SL . To find U we solve the following equation(37) U h + h e e e − h + h e e e − h − h U − = h +˜ h ˜ e ˜ e ˜ e − ˜ h +˜ h ˜ e ˜ e ˜ e − ˜ h − h . For the given permutation we find U (123) = q a a a a − q a a a a q a a a a q a a a a
00 0 1 . The 2 × U (123) in the n = 2 case. If U (123) actson a three dimensional space it represents a planar rotation over an angle θ , , .We will denote this matrix alternatively by R x x ( θ , , ) := U (123) . The x x indexrepresents the plane that is being rotated. This overlap can be represented by aswap as in Proposition 4.1:1121232 31234 4 Y A = { Q , Q } −→ Y A = { Q , Q } Example 4.6.
Consider in the algebra R ( h ) again the Lie algebra sl constructedfrom the labelling Abelian algebra Y := { Q , Q } . The second labelling Abelianalgebra we consider is Y := { Q , Q } . As before we can represent this overlapby a swap as in Proposition 4.1.1121232 31234 4 Y A = { Q , Q } −→
112 342 31234 4 Y A = { Q , Q } The generators corresponding to the second Abelian algebra are:˜ e := λ ([ Q , [ Q , Q ]] + a [ Q , Q ]) , ˜ e := λ ([ Q , [ Q , Q ]] − a [ Q , Q ]) , ˜ h := [ e , e ] , ˜ e := λ ([ Q , [ Q , Q ]] + a [ Q , Q ]) , ˜ e := λ ([ Q , [ Q , Q ]] − a [ Q , Q ]) , ˜ h := [ e , e ] , ˜ e := [ e , e ] , ˜ e := [ e , e ] . ACAH PROBLEMS FOR THE OSCILLATOR ALGEBRA 23
Solving equation (37) results in U Y Y := q a a a a − q a a a a q a a a a q a a a a . This is again an orthogonal matrix. Similarly to the previous example one can seethat the lower-right 2 × U Y Y representsa planar rotation matrtx over an angle π/ − θ , , but in a different plane. Wedenote this matrix by R x x ( θ , , − π/ . Given two labelling Abelian algebras, we are able to construct an isomorphismof sl and its corresponding rotation matrix. The previous examples show thatif one labelling Abelian algebra is obtained by a swap on the other, the resultingrotation matrices are planar. Twists on the other hand give reflections. This leadsto an alternative way to construct the rotation matrix of an isomorphism related totwo labelling Abelian algebras. We choose a path in the connection graph betweenthe vertices related to the labelling Abelian algebras. Every step along the pathin the connection graph can be represented by a swap as in Proposition 4.1. Eachof these swaps leads to a planar rotation. The final rotation matrix will be theproduct of each planar rotation found along the path. For example, consider thesetwo labelling Abelian algebras: h Q , Q i and h Q , Q i . The intermediate basesare the following: h Q , Q i → h Q , Q i → h Q , Q i → h Q , Q i . For each step we construct the corresponding rotation matrix. This results in R := R x x ( π/ − θ , , ) R x x ( − θ , , ) R x x ( θ , , − π/ . It is known that every rotation matrix R ∈ SO can be written as the followingproduct of planar rotations: R x x ( θ ) R x x ( θ ) R x x ( θ ) . The angles θ , θ and θ are the so-called Euler angles. By choosing the right paththrough the connection graph it is possible to give the decomposition of the rotationmatrix in planar matrices. The Euler angles also show up in the in the overlapcoefficients between the given bases. The overlap coefficients (34) are multivariateKrawtchouk polynomials of Griffiths type depending on three parameters R , , h , R , , h and R , , h . The relationship between these parameters and the angles is thefollowing: R K,L,Mh = cos(2 θ K,L,M ) . This relation between multivariate Krawtchouk polynomials and rotations had beendiscussed earlier in [24].We generalize the previous analysis to any n . To any pair of labelling Abelianalgebras we are able to construct overlap coefficients and an isomorphism of sl n − represented by a rotation matrix in SO n − . Any rotation matrix in SO n − can bewritten as the product of ( n − n − planar rotations. In the same way the overlapcoefficients are multivariable Krawtchouk polynomials of Griffiths type dependingon ( n − n − parameters. By Proposition 4.1 it takes at most ( n − n − steps through the connection graph. Each step provides us with a parameter and a planarrotation matrix leading to the right number of parameters and planar rotations.We wil showcase this with an example that also shows the link with Krawtchoukpolynomials of Tratnik type: Example 4.7.
Consider the labelling Abelian algebras Y initial := { Q [ k ] | < k < n } , Y final := { Q [ k...n ] | < k < n } . Between these two labelling Abelian algebras, it will take ( n − n − steps to findthe overlap.Step 1: Replace Q by Q in Y initial . The overlap coefficients are univariateKrawtchouk polynomials: B s s ( R , , h , C , , ) . The vector index s labels the basis diagonalized by Y initial . The index s labelsthe basis diagonalized by the new labelling Abelian algebra. The related rotationmatrix is given by R x x ( θ , , ) . Step 2: We perform two steps: Replace Q by Q and then Q by Q . Theoverlap coefficients are bivariate Krawtchouk polynomials of Tratnik type: B s s ( R , , h , C , , ) B s s ( R , , h , C , , ) . The related rotation matrix is given by R x x ( θ , , ) R x x ( θ , , ) . Step k −
1: Consider the first k − { Q k − k , Q [ k − ..k ] , . . . , Q [2 ..k ] , Q [ k ] } . We perform k − Q [ k ] by Q [2 ..k +1] . Next replace Q [2 ..k ] by Q [3 ..k +1] and so on until Q k − k is replaced by Q kk +1 . These k − k − k − Y l =1 B s k − l s kl ( R l, [ l +1 ..k ] ,k +1 h , C l, [ l +1 ..k ] ,k +1 ) . The related rotation matrix is given by k − Y l =1 R x k − l x k +1 − l ( θ l, [ l +1 ..k ] ,k +1 ) . Combining all n − B ~s ~s n − = n − X m =2 X s ml n − Y k =2 k − Y l =1 B s k − l s kl ( R l, [ l +1 ..k ] ,k +1 h , C l, [ l +1 ..k ] ,k +1 ) . The overlap coefficients are multivariate Krawtchouk polynomials of Griffiths typewhich are themselves a sum of products of Krawtchouk polynomials of Tratnik type.The overlap coefficients depend on ( n − n − parameters of the form R l, [ l +1 ..k ] ,k +1 h . ACAH PROBLEMS FOR THE OSCILLATOR ALGEBRA 25
The corresponding rotation matrix is given by a product of ( n − n − rotationmatrices. n − Y k =2 k − Y l =1 R x k +1 − l x k − l ( θ l, [ l +1 ..k ] ,k +1 )5. Conclusion
We introduced the oscillator Racah algebra R n ( h ). We have shown how toembed sl n − into the algebra R n ( h ). This connects the representation theory forboth algebras. In finite irreducible representations we considered bases diagonalizedby labelling Abelian algebras of R n ( h ). The overlap coefficients between a pair ofbases are shown to be multivariate Krawtchouk polynomials of Tratnik or Griffithstype. Isomorphisms of sl n − related to pairs of labelling Abelian algebras and theircorresponding Lie group elements were constructed and their link to the overlapcoefficients is explained. This has provided an explanation as to why the recouplingcoefficients of the oscillator algebra and the matrix elements of the restrictions toO( n −
1) of symmetric representations of SL( n −
1) are generically given in termsof the multivariate Krawtchouk polynomials of Griffiths.6.
Acknowledgements
NC is partially supported by Agence National de la Recherche Projet AHAANR-18-CE40-0001 and is gratefully holding a CRM-Simons professorship. WVDVthanks the Fonds Professor Frans Wuytack for supporting his research. He is alsograteful for the hospitality offered by him at the CRM during his stay. The re-search of LV is supported in part by a discovery grant of the Natural Science andEngineering Research Council (NSERC) of Canada.
Appendix A. Calculation of overlap coefficients
Let V be a finite dimensional representation of sl and ˜ . an automorphism of sl .The element h is a Cartan generator of sl . Let { ψ k } be an eigenbasis of h and { φ s } be an eigenbasis for ˜ h . The indices k and s run form 0 to N with dim( V ) = N + 1.We are interested in the overlap coefficients B ks between these bases:(38) φ s = N X k =0 B sk ψ k .hψ k = µ k ψ k ˜ hφ s = ν s φ s . The algebra sl has algebra relations [ h, e ] = 2 e and [ h, f ] = − f with e is theraising operator and f the lowering operator on { ψ k } : eψ k = e kk +1 ψ k +1 f ψ k = f kk − ψ k − and µ k = µ + 2 k . From the algebra relation [ e, f ] = h it follows that f kk − e k − k − e kk +1 f k +1 k = µ k . Let A k := e kk − f k − k . Then we have A k − A k +1 = 2 k + µ . From this we find A k = − k ( k − − µ k − Ω with Ω ∈ R . We express ˜ h as a linear combination of h , e and f .˜ h = R h h + R e e + R f f with R e R f + R h = 1. We have set up everything we need to find the overlapcoefficients. Let the operator ˜ h act on both sides of equality (38).˜ hψ s = N X k =0 B sk ( R h h + R e e + R f f ) ψ k . This gives ν s φ s = N X k =0 B sk ( R h µ k ψ k + R e e kk +1 ψ k +1 + R f f kk − ψ k − ) . We expand the left hand side into the basis ψ k and we gather the terms on theright hand side: N X k =0 ν s B sk ψ k = N X k =0 ( B sk R h µ k + B sk − R e e k − k + B sk +1 R f f k +1 k ) ψ k . From this we find the recurrence relation ν s B sk = B sk +1 R f f k +1 k + B sk R h µ k + B sk − R e e k − k . We want to recognize this recurrence relation as one of the family of orthogonalpolynomials. Let ˜ B ks = k Y t =2 f tt − R f ! B sk to find ν s ˜ B sk = ˜ B sk +1 + R h µ k ˜ B sk + R e R f e k − k f k − k ˜ B sk − . We write the coefficients as polynomials in x = ν s :(39) x ˜ B k ( x ) = ˜ B k +1 ( x ) + R h µ k ˜ B k ( x ) + R e R f A k ˜ B k − ( x ) . We want to compare this with the recurrence relation of the normalized Krawtchoukpolynomials as defined in [35] : xp n ( x ) = p n +1 + ( n (1 − r ) + rN ) p n ( x ) + r (1 − r ) n ( N + 1 − n ) p n − ( x )with n = 0 , , . . . , N . Let x = αy + β and introduce q n ( y ) = p n ( αy + β ) /α n . Thepolynomial q n ( x ) satisfies the following recurrence relation: yq n ( y ) = q n +1 + n (1 − r ) + rN − βα q n ( y ) + r (1 − r ) n ( N + 1 − n ) α q n − ( y ) . We retrieve equation (39) if we set α = 12 , r = 1 − R h , β = N , k = n, Ω = 0 , µ = − N. ACAH PROBLEMS FOR THE OSCILLATOR ALGEBRA 27
We explicitly write down the polynomials ˜ B k ( x ).˜ B k ( x ) = 2 k p k (cid:18) x + N (cid:19) = ( − N ) k (1 − R h ) k K k (cid:18) x + N − R h , N (cid:19) = ( − N ) k (1 − R h ) k F (cid:18) − k, − x + N − N | − R h (cid:19) . The overlap coefficients are the Krawtchouk polynomials K k ( x ) (defined in the lastline of the equation above) up to a normalization factor. B sk = k Y t =2 f tt − R f ! ( − N ) k (1 − R h ) k K k (cid:18) ν s + N − R h , N (cid:19) . References [1] P. Baseilhac. Deformed Dolan-Grady relations in quantum integrable models.
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