The Bannai-Ito algebra and some applications
Hendrik De Bie, Vincent X. Genest, Satoshi Tsujimoto, Luc Vinet, Alexei Zhedanov
aa r X i v : . [ m a t h - ph ] N ov The Bannai-Ito algebra and some applications
Hendrik De Bie
Department of Mathematical Analysis, Faculty of Engineering and Architecture, Ghent University, Galglaan 2Galglaan 2, 9000 Ghent, BelgiumE-mail: [email protected]
Vincent X. Genest
Centre de recherches mathématiques, Université de Montréal, P.O. Box 6128, Centre-ville Station, Montréal(QC) Canada, H3C 3J7E-mail: [email protected]
Satoshi Tsujimoto
Department of applied mathematics and physics, Kyoto University, Kyoto 6068501, JapanE-mail: [email protected]
Luc Vinet
Centre de recherches mathématiques, Université de Montréal, P.O. Box 6128, Centre-ville Station, Montréal(QC) Canada, H3C 3J7E-mail: [email protected]
Alexei Zhedanov
Donetsk Institute for Physics and Technology, Donetsk 340114, UkraineE-mail: [email protected]
Abstract.
The Bannai-Ito algebra is presented together with some of its applications. Its relations with theBannai-Ito polynomials, the Racah problem for the sl − (2) algebra, a superintegrable model with reflections anda Dirac-Dunkl equation on the 2-sphere are surveyed.
1. Introduction
Exploration through the exact solution of models has a secular tradition in mathematical physics.Empirically, exact solvability is possible in the presence of symmetries, which come in variousguises and which are described by a variety of mathematical structures. In many cases, exactsolutions are expressed in terms of special functions, whose properties encode the symmetries ofhe systems in which they arise. This can be represented by the following virtuous circle:Exact solvabilitySymmetries x x Special functions ' ' k k Algebraic structures s s * * f f The classical path is the following: start with a model, find its symmetries, determine how thesesymmetries are mathematically described, work out the representations of that mathematicalstructure and obtain its relation to special functions to arrive at the solution of the model.However, one can profitably start from any node on this circle. For instance, one can identifyand characterize new special functions, determine the algebraic structure they encode, look formodels that have this structure as symmetry algebra and proceed to the solution. In this paper,the following path will be taken:Algebra −→ Orthogonal polynomials −→ Symmetries −→ Exact solutionsThe outline of the paper is as follows. In section 2, the Bannai-Ito algebra is introduced andsome of its special cases are presented. In section 3, a realization of the Bannai-Ito algebra interms of discrete shift and reflection operators is exhibited. The Bannai-Ito polynomials andtheir properties are discussed in section 4. In section 5, the Bannai-Ito algebra is used to derivethe recurrence relation satisfied by the Bannai-Ito polynomials. In section 6, the parabosonalgebra and the sl − (2) algebra are introduced. In section 7, the realization of sl − (2) in termsof Dunkl operators is discussed. In section 8, the Racah problem for sl − (2) and its relation withthe Bannai-Ito algebra is examined. A superintegrable model on the 2-sphere with Bannai-Itosymmetry is studied in section 9. In section 10, a Dunkl-Dirac equation on the 2-sphere withBannai-Ito symmetry is discussed. A list of open questions is provided in lieu of a conclusion.
2. The Bannai-Ito algebra
Throughout the paper, the notation [ A , B ] = AB − BA and { A , B } = AB + BA will be used. Let ω , ω and ω be real parameters. The Bannai-Ito algebra is the associative algebra generated by K , K and K together with the three relations { K , K } = K + ω , { K , K } = K + ω , { K , K } = K + ω , (1)or { K i , K j } = K k + ω k , with ( i jk ) a cyclic permutation of (1, 2, 3). The Casimir operator Q = K + K + K ,commutes with every generator; this property is easily verified with the commutator identity[ AB , C ] = A { B , C } − { A , C } B . Let us point out two special cases of (1) that have been consideredpreviously in the literature.(i) ω = ω = ω = { K , K } = K , { K , K } = K , { K , K } = K ,is sometimes referred to as the anticommutator spin algebra [1, 2]; representations of this algebrawere examined in [1, 2, 3, 4].ii) ω = ω = ω In recent work on the construction of novel finite oscillator models [5, 6], E. Jafarov, N. Stoilovaand J. Van der Jeugt introduced the following extension of u (2) by an involution R ( R = I , R ] = { I , R } = { I , R } = I , I ] = iI , [ I , I ] = iI , [ I , I ] = i ( I + ω R ).It is easy to check that with K = iI R , K = I , K = I R ,the above relations are converted into { K , K } = K , { K , K } = K , { K , K } = K + ω .
3. A realization of the Bannai-Ito algebra with shift and reflections operators
Let T + and R be defined as follows: T + f ( x ) = f ( x + R f ( x ) = f ( − x ).Consider the operator b K = F ( x )(1 − R ) + G ( x )( T + R − + h , h = ρ + ρ − r − r + F ( x ) and G ( x ) given by F ( x ) = ( x − ρ )( x − ρ ) x , G ( x ) = ( x − r + x − r + x + ρ , ρ , r , r are four real parameters. It can be shown that b K is the most general operatorof first order in T + and R that stabilizes the space of polynomials of a given degree [7]. That is,for any polynomial Q n ( x ) of degree n , [ b K Q n ( x )] is also a polynomial of degree n . Introduce b K = x + x ” operator and b K ≡ { b K , b K } − ρ ρ − r r ). (4)It is directly verified that b K , b K and b K satisfy the commutation relations { b K , b K } = b K + b ω , { b K , b K } = b K + b ω , { b K , b K } = b K + b ω , (5)where the structure constants b ω , b ω and b ω read b ω = ρ ρ + r r ), b ω = ρ + ρ − r − r ), b ω = ρ ρ − r r ). (6)The operators b K , b K and b K thus realize the Bannai-Ito algebra. In this realization, the Casimiroperator acts as a multiple of the identity; one has indeed b Q = b K + b K + b K = ρ + ρ + r + r ) − . The Bannai-Ito polynomials Since the operator (2) preserves the space of polynomials of a given degree, it is natural to lookfor its eigenpolynomials, denoted by B n ( x ), and their corresponding eigenvalues λ n . We use thefollowing notation for the generalized hypergeometric series [8] r F s µ a , . . ., a r b , . . ., b s ¯¯¯ z ¶ = ∞ X k = ( a ) k · · · ( a r ) k ( b ) k · · · ( b s ) k z k k ! ,where ( c ) k = c ( c + · · · ( c + k − c ) ≡ a i is a negative integer. Solving the eigenvalue equation b K B n ( x ) = λ n B n ( x ), n =
0, 1, 2, . . . (7)it is found that the eigenvalues λ n are given by [7] λ n = ( − n ( n + h ), (8)and that the polynomials have the expression B n ( x ) c n = F µ − n , n + + h , x − r + − x − r + − r − r , ρ − r + , ρ − r + ¯¯¯ ¶ + ( n )( x − r + )( ρ − r + )( ρ − r + ) 4 F µ − n , n + + h , x − r + − x − r + − r − r , ρ − r + , ρ − r + ¯¯¯ ¶ n even, F µ − n − , n + h , x − r + , − x − r + − r − r , ρ − r + , ρ − r + ¯¯¯ ¶ − ( n + h )( x − r + )( ρ − r + )( ρ − r + ) 4 F µ − n − , n + + h , x − r + , − x − r + − r − r , ρ − r + , ρ − r + ¯¯¯ ¶ n odd, (9)where the coefficient c n + p = ( − p (1 − r − r ) n ( ρ − r + ρ − r + n + p ( n + h + n + p , p ∈ {
0, 1 } ,ensures that the polynomials B n ( x ) are monic, i.e. B n ( x ) = x n + O ( x n − ). The polynomials (9)were first written down by Bannai and Ito in their classification of the orthogonal polynomialssatisfying the Leonard duality property [9, 10], i.e. polynomials p n ( x ) satisfying both • A 3-term recurrence relation with respect to the degree n , • A 3-term difference equation with respect to a variable index s .The identification of the defining eigenvalue equation (7) of the Bannai-Ito polynomials in [7] hasallowed to develop their theory. That they obey a three-term difference equation stems from thefact that there are grids such as x s = ( − s ( s /2 + a + − H = A ( x ) R + B ( x ) T + R + C ( x ),re tridiagonal in the basis f ( x s ) H f ( x s ) = ( B ( x s ) f ( x s + ) + A ( x s ) f ( x s − ) + C ( x s ) f ( x s ) s even, A ( x s ) f ( x s + ) + B ( x s ) f ( x s − ) + C ( x s ) f ( x s ) s odd.It was observed by Bannai and Ito that the polynomials (9) correspond to a q → − q -Racah polynomials (see [11] for the definition of q -Racah polynomials). In this connection, it isworth mentioning that the Bannai-Ito algebra (5) generated by the defining operator b K and therecurrence operator b K of the Bannai-Ito polynomials can be obtained as a q → − q -Racah polynomials. TheBannai-Ito polynomials B n ( x ) have companions I n ( x ) = B n + ( x ) − B n + ( ρ ) B n ( ρ ) B n ( x ) x − ρ ,called the complementary Bannai-Ito polynomials [13]. It has now been understood that thepolynomials B n ( x ) and I n ( x ) are the ancestors of a rich ensemble of polynomials referred toas “ −
5. The recurrence relation of the BI polynomials from the BI algebra
Let us now show how the Bannai-Ito algebra can be employed to derive the recurrence relationsatisfied by the Bannai-Ito polynomials. In order to obtain this relation, one needs to find theaction of the operator b K on the BI polynomials B n ( x ). Introduce the operators b K + = ( b K + b K )( b K − − b ω + b ω b K − = ( b K − b K )( b K + + b ω − b ω b K i and b ω i are given by (2), (3), (4) and (6). It is readily checked using (5) that { b K , b K ± } = ± K ± .One can directly verify that b K ± maps polynomials to polynomials. In view of the above, one has b K b K + B n ( x ) = ( − b K + b K + b K + ) B n ( x ) = (1 − λ n ) b K + B n ( x ),where λ n is given by (8). It is also seen from (8) that1 − λ n = ( λ n − n even, λ n + n odd.It follows that b K + B n ( x ) = ( α (0) n B n − ( x ) n even, α (1) n B n + ( x ) n odd.Similarly, one finds b K − B n ( x ) = ( β (0) n B n + ( x ) n even, β (1) n B n − ( x ) n odd.he coefficients α (0) n = n ( n + ρ + ρ )( r + r − n )( n − + h ) n + h − , α (1) n = − n + h + β (0) n = n + h + β (1) n = ρ − r + n )( ρ − r + n )( ρ − r + n )( ρ − r + n ) n + h − V = b K + ( b K + + b K − ( b K − b K ± , it follows that V = b K ( b K − − b ω b K − b ω /2. (12)From (7), (11) and the actions of the operators b K ± , we find that V is two-diagonal V B n ( x ) = ( ( λ n + α (0) n B n − ( x ) + ( λ n − β (0) n B n + ( x ) n even,( λ n − β (1) n B n − ( x ) + ( λ n + α (1) n B n + ( x ) n odd. (13)From (12) and recalling the definition (3) of b K , we have also V B n ( x ) = £ ( λ n − x + − b ω λ n − b ω /2 ¤ B n ( x ). (14)Upon combining (13) and (14), one finds that the Bannai-Ito polynomials satisfy the three-termrecurrence relation x B n ( x ) = B n + ( x ) + ( ρ − A n − C n ) B n ( x ) + A n − C n B n − ( x ),where A n = ( ( n + + ρ − r )( n + + ρ − r )4( n + ρ + ρ − r − r + n even, ( n + + ρ + ρ − r − r )( n + + ρ + ρ )4( n + ρ + ρ − r − r + n odd, C n = ( − n ( n − r − r )4( n + ρ + ρ − r − r ) n even, − ( n + ρ − r )( n + ρ − r )4( n + ρ + ρ − r − r ) n odd. (15)The positivity of the coefficient A n − C n restricts the polynomials B n ( x ) to being orthogonal on afinite set of points [19].
6. The paraboson algebra and sl − (2)The next realization of the Bannai-Ito algebra will involve sl − (2); this algebra, introduced in[20], is closely related to the parabosonic oscillator. Let a and a † be the generators of the paraboson algebra. These generators satisfy [21][ { a , a † } , a ] = − a , [ { a , a † } , a † ] = a † .Setting H = { a , a † } , the above relations amount to[ H , a ] = − a , [ H , a † ] = a † ,which correspond to the quantum mechanical equations of an oscillator. .2. Relation with osp (1 | osp (1 |
2) [22]. Indeed, upon setting F − = a , F + = a † , E = H = { F + , F − } , E + = F + , E − = F − ,and interpreting F ± as odd generators, it is directly verified that the generators F ± , E ± and E satisfy the defining relations of osp (1 |
2) [23]:[ E , F ± ] = ± F ± , { F + , F − } = E , [ E , E ± ] = ± E ± , [ E − , E + ] = E ,[ F ± , E ± ] =
0, [ F ± , E ∓ ] = ∓ F ∓ .The osp (1 |
2) Casimir operator reads C osp (1 | = ( E − − E + E − − F + F − . q (2)Consider now the quantum algebra sl q (2). It can be presented in terms of the generators A and A ± satisfying the commutation relations [24][ A , A ± ] = ± A ± , [ A − , A + ] = q A − q − A q − q − .Upon setting B + = A + q ( A − , B − = q ( A − A − , B = A ,these relations become[ B , B ± ] = ± B ± , B − B + − qB + B − = q B − q − sl q (2) Casimir operator is of the form C sl q (2) = B + B − q − B − q − q −
1) ( q B − + q − B ).Let j be a non-negative integer. The algebra sl q (2) admits a discrete series representation on thebasis | j , n 〉 with the actions q B | j , n 〉 = q j + n | j , n 〉 , n =
0, 1, 2, . . ..The algebra has a non-trivial coproduct ∆ : sl q (2) → sl q (2) ⊗ sl q (2) which reads ∆ ( B ) = B ⊗ + ⊗ B , ∆ ( B ± ) = B ± ⊗ q B + ⊗ B ± . − (2) algebra as a q → − limit of sl q (2)The sl − (2) algebra can be obtained as a q → − sl q (2). Let us first introduce the operator R defined as R = lim q →− q B .t is easily seen that R | j , n 〉 = ( − j + n | j , n 〉 = ǫ ( − n | j , n 〉 ,where ǫ = ± j , thus R =
1. When q → −
1, one finds that q B B + = qB + q B B − q B = qq B B − −→ { R , B ± } = B − B + − qB + B − = q B − q − −→ { B + , B − } = B , C sl q (2) −→ B + B − R − B R + R /2, ∆ ( B ± ) = B ± ⊗ q B + ⊗ B ± −→ ∆ ( B ± ) = B ± ⊗ R + ⊗ B ± .In summary, sl − (2) is the algebra generated by J , J ± and R with the relations [20][ J , J ± ] = ± J ± , [ J , R ] = { J ± , R } = { J + , J − } = J , R =
1. (16)The Casimir operator has the expression Q = J + J − R − J R + R /2, (17)and the coproduct is of the form [25] ∆ ( J ) = J ⊗ + ⊗ J , ∆ ( J ± ) = J ± ⊗ R + ⊗ J ± , ∆ ( R ) = R ⊗ R . (18)The sl − (2) algebra (16) has irreducible and unitary discrete series representations with basis | ǫ , µ ; n 〉 , where n is a non-negative integer, ǫ = ± µ is a real number such that µ > − J | ǫ , µ ; n 〉 = ( n + µ +
12 ) | ǫ , µ ; n 〉 , R | ǫ , µ ; n 〉 = ǫ ( − n | ǫ , µ ; n 〉 , J + | ǫ , µ ; n 〉 = ρ n + | ǫ , µ ; n + 〉 , J − | ǫ , µ ; n 〉 = ρ n | ǫ , µ ; n − 〉 ,where ρ n = p n + µ (1 − ( − n ). In these representations, the Casimir operator takes the value Q | ǫ , µ ; n 〉 = − ǫµ | ǫ , µ ; n 〉 .These modules will be denoted by V ( ǫ , µ ) . Let us offer the following remarks. • The sl − (2) algebra corresponds to the parabose algebra supplemented by R . • The sl − (2) algebra consists of the Cartan generator J and the two odd elements of osp (1 | R . • One has C osp (1 | = Q , where Q is given by (17). Thus the introduction of R allows to takethe square-root of C osp (1 | . • In sl − (2), one has [ J − , J + ] = − QR . On the module V ( ǫ , µ ) , this leads to[ J − , J + ] = + ǫµ R . . Dunkl operators The irreducible modules V ( ǫ , µ ) of sl − (2) can be realized by Dunkl operators on the real line. Let R x be the reflection operator R x f ( x ) = f ( − x ).The Z -Dunkl operator on R is defined by [26] D x = ∂∂ x + ν x (1 − R x ),where ν is a real number such that ν > − b J ± = p x ∓ D x ),and defining b J = { b J − , b J + } , it is readily verified that a realization of the sl − (2)-module V ( ǫ , µ ) with ǫ = µ = ν is obtained. In particular, one has[ b J − , b J + ] = + ν R x .It can be seen that b J † ± = b J ∓ with respect to the measure | x | ν d x on the real line [27].
8. The Racah problem for sl − (2) and the Bannai-Ito algebra The Racah problem for sl − (2) presents itself when the direct product of three irreduciblerepresentations is examined. We consider the three-fold tensor product V = V ( ǫ , µ ) ⊗ V ( ǫ , µ ) ⊗ V ( ǫ , µ ) .It follows from the coproduct formula (18) that the generators of sl − (2) on V are of the form J (4) = J (1)0 + J (2)0 + J (3)0 , J (4) ± = J (1) ± R (2) R (3) + J (2) ± R (3) + J (3) ± , R (4) = R (1) R (2) R (3) ,where the superscripts indicate on which module the generators act. In forming the module V ,two sequences are possible: one can first combine (1) and (2) to bring (3) after or one can combine(2) and (3) before adding (1). This is represented by ³ V ( ǫ , µ ) ⊗ V ( ǫ , µ ) ´ ⊗ V ( ǫ , µ ) or V ( ǫ , µ ) ⊗ ³ V ( ǫ , µ ) ⊗ V ( ǫ , µ ) ´ . (19)These two addition schemes are equivalent and the two corresponding bases are unitarily related.In the following, three types of Casimir operators will be distinguished. • The initial Casimir operators Q i = J ( i ) + J ( i ) − R ( i ) − ( J ( i )0 − R ( i ) = − ǫ i µ i , i =
1, 2, 3. • The intermediate Casimir operators Q i j = ( J ( i ) + R ( j ) + J ( j ) + )( J ( i ) − R ( j ) + J ( j ) − ) R ( i ) R ( j ) − ( J ( i )0 + J ( j )0 − R ( i ) R ( j ) = ( J ( i ) − J ( j ) + − J ( i ) + J ( j ) − ) R ( i ) − R ( i ) R ( j ) /2 + Q i R ( j ) + Q j R ( i ) ,where ( i j ) = (12), (23). The total Casimir operator Q = [ J (4) + J (4) − − ( J (4)0 − R (4) .Let | q , q ; m 〉 and | q , q ; m 〉 be the orthonormal bases associated to the two coupling schemespresented in (19). These two bases are defined by the relations Q | q , q ; m 〉 = q | q , q ; m 〉 , Q | q , q ; m 〉 = q | q , q ; m 〉 ,and Q |− , q ; m 〉 = q |− , q ; m 〉 , J (4)0 |− , q ; m 〉 = ( m + µ + µ + µ + |− , q ; m 〉 .The Racah problem consists in finding the overlap coefficients 〈 q , q | q , q 〉 ,between the eigenbases of Q and Q with a fixed value q of the total Casimir operator Q ; asthese coefficients do not depend on m , we drop this label. For simplicity, let us now take ǫ = ǫ = ǫ = K = − Q , K = − Q ,one finds that the intermediate Casimir operators of sl − (2) realize the Bannai-Ito algebra [28] { K , K } = K + Ω , { K , K } = K + Ω , { K , K } = K + Ω , (20)with structure constants Ω = µ µ + µ µ ), Ω = µ µ + µ µ ), Ω = µ µ + µ µ ), (21)where µ = ǫ µ = − q . The first relation in (20) can be taken to define K which reads K = ( J (1) + J (3) − − J (1) − J (3) + ) R (1) R (2) + R (1) R (3) /2 − Q R (3) − Q R (1) .In the present realization the Casimir operator of the Bannai-Ito algebra becomes Q BI = µ + µ + µ + µ − V with basis vectors defined as eigenvectors of Q or of Q . The first step is to obtain the spectra of the intermediate Casimir operators. Simpleconsiderations based on the nature of the sl − (2) representation show that the eigenvalues q and q of Q and Q take the form [29, 30, 28, 20]: q = ( − s + ( s + µ + µ + q = ( − s ( s + µ + µ + s , s =
0, 1, . . ., N . The non-negative integer N is specified by N + = µ − µ − µ − µ .enote the eigenstates of K by | k 〉 and those of K by | s 〉 ; one has K | k 〉 = ( − k ( k + µ + µ + | k 〉 , K | s 〉 = ( − s ( s + µ + µ + | s 〉 .Given the expressions (21) for the structure constants Ω k , one can proceed to determine the( N + × ( N +
1) matrices that verify the anticommutation relations (20). The action of K on | k 〉 is found to be [28]: K | k 〉 = U k + | k + 〉 + V k | k 〉 + U k | k − 〉 ,with V k = µ + µ + − B k − D k and U k = p B k − D k where B k = ( ( k + µ + k + µ + µ + µ − µ + k + µ + µ + k even, ( k + µ + µ + k + µ + µ + µ + µ + k + µ + µ + k odd, D k = ( − k ( k + µ + µ − µ − µ )2( k + µ + µ )2) n even, − ( k + µ )( k + µ + µ − µ + µ )2( k + µ + µ ) n odd.Under the identifications ρ =
12 ( µ + µ ), ρ =
12 ( µ + µ ), r =
12 ( µ − µ ), r =
12 ( µ − µ ),one has B k = A k , D k = C k , where A k and C k are the recurrence coefficients (15) of the Bannai-Ito polynomials. Upon setting 〈 s | k 〉 = w ( s )2 k B k ( x s ), B ( x s ) ≡ 〈 s | K | k 〉 = ( − s ( s + ρ + 〈 s | k 〉 ,and on the other hand 〈 s | K | k 〉 = U k + 〈 s | k + 〉 + V k 〈 s | k 〉 + U k − 〈 s | k − 〉 .Comparing the two RHS yields x s B k ( x s ) = B k + ( x s ) + ( ρ − A k − C k ) B k ( x s ) + A k − C k B k − ( x s ),where x s are the points of the Bannai-Ito grid x s = ( − s ³ s + ρ + ´ − s =
0, . . ., N .Hence the Racah coefficients of sl − (2) are proportional to the Bannai-Ito polynomials. Thealgebra (20) with structure constants (21) is invariant under the cyclic permutations of the pairs( K i , µ i ). As a result, the representations in the basis where K is diagonal can be obtained directly.In this basis, the operator K is seen to be tridiagonal, which proves again that the Bannai-Itopolynomials possess the Leonard duality property. . A superintegrable model on S with Bannai-Ito symmetry We shall now use the analysis of the Racah problem for sl − (2) and its realization in terms ofDunkl operators to obtain a superintegrable model on the two-sphere. Recall that a quantumsystem in n dimensions with Hamiltonian H is maximally superintegrable it it possesses 2 n − H [31]. Let( s , s , s ) ∈ R and take s + s + s =
1. The standard angular momentum operators are L = i µ s ∂∂ s − s ∂∂ s ¶ , L = i µ s ∂∂ s − s ∂∂ s ¶ , L = i µ s ∂∂ s − s ∂∂ s ¶ .The system governed by the Hamiltonian H = L + L + L + µ s ( µ − R ) + µ s ( µ − R ) + µ s ( µ − R ), (22)with µ i , i =
1, 2, 3, real parameters such that µ i > − R i reflect the variable s i : R i f ( s i ) = f ( − s i ).(ii) The operators R i commute with the Hamiltonian: [ H , R i ] = R i by κ i = ±
1. This then treats the 8 potential terms µ s ( µ − κ ) + µ s ( µ − κ ) + µ s ( µ − κ ),simultaneously much like supersymmetric partners.(iv) Rescaling s i → rs i and taking the limit as r → ∞ gives the Hamiltonian of the Dunkloscillator [27, 33] e H = − [ D x + D x ] + b µ ( x + x ),after appropriate renormalization; see also [34, 35, 36].It can be checked that the following three quantities commute with the Hamiltonian (22) [29, 32]: C = µ iL + µ s s R − µ s s R ¶ R + µ R + µ R + R R /2, C = µ − iL + µ s s R − µ s s R ¶ R R + µ R + µ R + R R /2, C = µ iL + µ s s R − µ s s R ¶ R + µ R + µ R + R R /2,that is, [ H , C i ] = i =
1, 2, 3. To determine the symmetry algebra generated by the aboveconstants of motion, let us return to the Racah problem for sl − (2). Consider the following (gaugetransformed) parabosonic realization of sl − (2) in the three variables s i : J ( i ) ± = p · s i ∓ ∂ s i ± µ i s i R i ¸ , J ( i )0 = " − ∂ s i + s i + µ i s i ( µ i − R i ) , R ( i ) = R i , (23)for i =
1, 2, 3. Consider also the addition of these three realizations so that J = J (1)0 + J (2)0 + J (3)0 , J ± = J (1) ± R (2) R (3) + J (2) ± R (3) + J (3) ± , R = R (1) R (2) R (3) . (24)t is observed that in the realization (24), the total Casimir operator can be expressed in terms ofthe constants of motion as follows: Q = − C R (1) − C R (2) − C R (3) + µ R (2) R (3) + µ R (1) R (3) + µ R (1) R (2) + R /2,Upon taking Ω = QR ,one finds Ω + Ω = L + L + L + ( s + s + s ) Ã µ s ( µ − R ) + µ s ( µ − R ) + µ s ( µ − R ) ! , (25)so that H = Ω + Ω if s + s + s =
1. Assuming this constraint can be imposed, H is a quadraticcombination of QR . By construction, the intermediate Casimir operators Q i j commute with thetotal Casimir operator Q and with R and hence with Ω ; they thus commute with H = Ω + Ω andare the constants of motion. It is indeed found that Q = − C , Q = − C ,in the parabosonic realization (23). Let us return to the constraint s + s + s =
1. Observe that12 ( J + + J − ) = ( s R R + s R + s ) = s + s + s .Because ( J + + J − ) commutes with Ω = QR , Q and Q , one can impose s + s + s =
1. Sinceit is already known that the intermediate Casimir operators in the addition of three sl − (2)representations satisfy the Bannai-Ito structure relations, the constants of motion verify { C , C } = C − µ Q + µ µ , { C , C } = C − µ Q + µ µ , { C , C } = C − µ Q + µ µ ,and thus the symmetry algebra of the superintegrable system with Hamiltonian (22) is a centralextension (with Q begin the central operator) of the Bannai-Ito algebra. Let us note that therelation H = Ω + Ω relates to chiral supersymmetry since with S = Ω + { S , S } = H +
10. A Dunkl-Dirac equation on S Consider the Z -Dunkl operators D i = ∂∂ x i + µ i x i (1 − R i ), i =
1, 2, . . ., n ,with µ i > − Z n -Dunkl-Laplace operator is ~ D = n X i = D i .ith γ n the generators of the Euclidean Clifford algebra { γ m , γ n } = δ nm ,the Dunkl-Dirac operator is/ D = n X i = γ i D i .Clearly, one has / D = ~ D . Let us consider the three-dimensional case. Introduce the Dunkl“angular momentum” operators J = i ( x D − x D ), J = i ( x D − x D ), J = i ( x D − x D ).Their commutation relations are found to be[ J i , J k ] = i ǫ jkl J l (1 + µ l R l ). (26)The Dunkl-Laplace equation separates in spherical coordinates; i.e. one can write ~ D = D + D + D = M r + r ∆ S ,where ∆ S is the Dunkl-Laplacian on the 2-sphere. It can be verified that [36] ~ J = J + J + J = − ∆ S + µ µ (1 − R R ) + µ µ (1 − R R ) + µ µ (1 − R R ) − µ R − µ R − µ R + µ + µ + µ . (27)In three dimensions the Euclidean Clifford algebra is realized by the Pauli matrices σ = µ ¶ , σ = µ − ii ¶ , σ = µ − ¶ ,which satisfy σ i σ j = i ǫ i jk σ k + δ i j .Consider the following operator: Γ = ( ~ σ · ~ J ) + ~ µ · ~ R ,with ~ µ · ~ R = µ R + µ R + µ R . Using the commutation relations (26) and the expression (27) for ~ J , it follows that Γ + Γ = − ∆ S + ( µ + µ + µ )( µ + µ + µ + sl − (2) Casimir operator. This justifies calling Γ a Dunkl-Dirac operator on S sincea quadratic expression in Γ gives ∆ S . The symmetries of Γ can be constructed. They are found tohave the expression [37] M i = J i + σ i ( µ j R j + µ k R k + i jk ) cyclic,nd one has [ Γ , M i ] =
0. It is seen that the operators X i = σ i R i i =
1, 2, 3also commute with Γ . Furthermore, one has[ M i , X i ] = { M i , X j } = { M i , X k } = Y = − iX X X = R R R is central (like Γ ). The commutation relations satisfied bythe operators M i are[ M i , M j ] = i ǫ i jk ¡ M k + µ k ( Γ + X k ¢ + µ i µ j [ X i , X j ].This is again an extension of su (2) with reflections and central elements. Let K i = M i X i Y = M i σ i R j R k .It is readily verified that the operators K i satisfy { K , K } = K + µ ( Γ + Y + µ µ , { K , K } = K + µ ( Γ + Y + µ µ , { K , K } = K + µ ( Γ + Y + µ µ ,showing that the Bannai-Ito algebra is a symmetry subalgebra of the Dunkl-Dirac equation on S .Therefore, the Bannai-Ito algebra is also a symmetry subalgebra of the Dunkl-Laplace equation.
11. Conclusion
In this paper, we have presented the Bannai-Ito algebra together with some of its applications. Inconcluding this overview, we identify some open questions.(i) Representation theory of the Bannai-Ito algebraFinite-dimensional representations of the Bannai-Ito algebra associated to certain modelswere presented. However, the complete characterization of all representations of the Bannai-Ito algebra is not known.(ii) SupersymmetryThe parallel with supersymmetry has been underscored at various points. One may wonderif there is a deeper connection.(iii) Dimensional reductionIt is well known that quantum superintegrable models can be obtained by dimensionalreduction. It would be of interest to adapt this framework in the presence of reflectionsoperators. Could the BI algebra can be interpreted as a W -algebra ?(iv) Higher ranksOf great interest is the extension of the Bannai-Ito algebra to higher ranks, in particular formany-body applications. In this connection, it can be expected that the symmetry analysis ofhigher dimensional superintegrable models or Dunkl-Dirac equations will be revealing. Acknowledgements
V.X.G. holds an Alexander-Graham-Bell fellowship from the Natural Science and EngineeringResearch Council of Canada (NSERC). The research of L.V. is supported in part by NSERC. H.DB. and A.Z. have benefited from the hospitality of the Centre de recherches mathématiques(CRM). eferences [1] Arik M and Kayserilioglu U 2003
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