The Bannai-Ito algebra and some applications

Journal of Physics: Conference Series PAPER OPEN ACCESS The Bannai-Ito algebra and some applications To cite this article: Hendrik De Bie et al 015 J. Phys.: Conf. Ser. 597 01001 View the article online for updates and enhancements. Related content - The Bannai--Ito algebra and a superintegrable system with reflections on the two-sphere Vincent X Genest, Luc Vinet and Alexei Zhedanov - A finite oscillator model with equidistant position spectrum based on an extension of ${\mathfrak{su}}$ Roy Oste and Joris Van der Jeugt - A superintegrable model with reflections on Sn1 and the higher rank Bannai Ito algebra Hendrik De Bie, Vincent X Genest, Jean- Michel Lemay et al. Recent citations - The dual pair Pinn osp1, the Dirac equation and the Bannai-Ito algebra Julien Gaboriaud et al - The q-onsager algebra and multivariable q-special functions Pascal Baseilhac et al - A Dirac Dunkl Equation on S and the Bannai Ito Algebra Hendrik De Bie et al This content was downloaded from IP address 148.51.3.83 on 30/10/018 at 17:04

30th International Colloquium on Group Theoretical Methods in Physics Group30 The Bannai-Ito algebra and some applications Hendrik De Bie 1, Vincent X Genest, Satoshi Tsujimoto 3, Luc Vinet and Alexei Zhedanov 4 1 Department of Mathematical Analysis, Faculty of Engineering and Architecture, Ghent University, Galglaan Galglaan, 9000 Ghent, Belgium Centre de recherches mathématiques, Université de Montréal, P.O. Box 618, Centre-ville Station, Montréal QC Canada, H3C 3J7 3 Department of applied mathematics and physics, Kyoto University, Kyoto 6068501, Japan 4 Donetsk Institute for Physics and Technology, Donetsk 340114, Ukraine E-mail: hendrik.debie@ugent.be, vincent.genest@umontreal.ca, tujimoto@i.kyoto-u.ac.jp, luc.vinet@umontreal.ca and zhedanov@yahoo.com Abstract. The Bannai-Ito algebra is presented together with some of its applications. Its relations with the Bannai-Ito polynomials, the Racah problem for the sl 1 algebra, a superintegrable model with reflections and a Dirac-Dunkl equation on the -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 various guises and which are described by a variety of mathematical structures. In many cases, exact solutions are expressed in terms of special functions, whose properties encode the symmetries of the systems in which they arise. This can be represented by the following virtuous circle: Exact solvability Symmetries Special functions Algebraic structures The classical path is the following: start with a model, find its symmetries, determine how these symmetries are mathematically described, work out the representations of that mathematical structure 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 identify and characterize new special functions, determine the algebraic structure they encode, look for models 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 solutions Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the authors and the title of the work, journal citation and DOI. Published under licence by Ltd 1

The outline of the paper is as follows. In section, the Bannai-Ito algebra is introduced and some of its special cases are presented. In section 3, a realization of the Bannai-Ito algebra in terms of discrete shift and reflection operators is exhibited. The Bannai-Ito polynomials and their properties are discussed in section 4. In section 5, the Bannai-Ito algebra is used to derive the recurrence relation satisfied by the Bannai-Ito polynomials. In section 6, the paraboson algebra and the sl 1 algebra are introduced. In section 7, the Racah problem for sl 1 and its relation with the Bannai-Ito algebra is examined. A superintegrable model on the -sphere with Bannai-Ito symmetry is studied in section 8. In section 9, a Dunkl-Dirac equation on the -sphere with Bannai-Ito symmetry is discussed. A list of open questions is provided in lieu of a conclusion.. The Bannai-Ito algebra Throughout the paper, the notation [A, B] = AB BA and {A, B} = AB +BA will be used. Let ω 1, ω and ω 3 be real parameters. The Bannai-Ito algebra is the associative algebra generated by K 1, K and K 3 together with the three relations {K 1, K } = K 3 + ω 3, {K, K 3 } = K 1 + ω 1, {K 3, K 1 } = K + ω, 1 or {K i, K j } = K k + ω k, with ijk a cyclic permutation of 1,, 3. The Casimir operator Q = K 1 + K + K 3, 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 considered previously in the literature. i ω 1 = ω = ω 3 = 0 The special case with defining relations {K 1, K } = K 3, {K, K 3 } = K 1, {K 3, K 1 } = K, is sometimes referred to as the anticommutator spin algebra [1, ]; representations of this algebra were examined in [1,, 3, 4]. ii ω 1 = ω = 0 ω 3 In recent work on the construction of novel finite oscillator models [5, 6], E. Jafarov, N. Stoilova and J. Van der Jeugt introduced the following extension of u by an involution R R = 1: It is easy to check that with the above relations are converted into [I 3, R] = 0, {I 1, R} = 0, {I, R} = 0, [I 3, I 1 ] = ii, [I, I 3 ] = ii 1, [I 1, I ] = ii 3 + ω 3 R. K 1 = ii 1 R, K = I, K 3 = I 3 R, {K 1, K 3 } = K, {K, K 3 } = K 1, {K 1, K } = K 3 + ω 3.

3. A realization of the Bannai-Ito algebra with shift and reflections operators Let T + and R be defined as follows: T + fx = fx + 1, Rfx = f x. Consider the operator K 1 = F x1 R + GxT + R 1 + h, h = ρ 1 + ρ r 1 r + 1/, with F x and Gx given by F x = x ρ 1x ρ x, Gx = x r 1 + 1/x r + 1/, x + 1/ where ρ 1, ρ, r 1, r are four real parameters. It can be shown that K 1 is the most general operator of 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, [ K 1 Q n x] is also a polynomial of degree n. Introduce which is essentially the multiplication by x operator and K = x + 1/, 3 K 3 { K 1, K } 4ρ 1 ρ r 1 r. 4 It is directly verified that K 1, K and K 3 satisfy the commutation relations { K 1, K } = K 3 + ω 3, { K, K 3 } = K 1 + ω 1, { K 3, K 1 } = K + ω, 5 where the structure constants ω 1, ω and ω 3 read ω 1 = 4ρ 1 ρ + r 1 r, ω = ρ 1 + ρ r 1 r, ω 3 = 4ρ 1 ρ r 1 r. 6 The operators K 1, K and K 3 thus realize the Bannai-Ito algebra. Casimir operator acts as a multiple of the identity; one has indeed In this realization, the Q = K 1 + K + K 3 = ρ 1 + ρ + r 1 + r 1/4. 4. The Bannai-Ito polynomials Since the operator preserves the space of polynomials of a given degree, it is natural to look for its eigenpolynomials, denoted by B n x, and their corresponding eigenvalues λ n. We use the following notation for the generalized hypergeometric series [8] a1,..., a r rf s z = b 1,..., b s k=0 a 1 k a r k z k b 1 k b s k k!, where c k = cc + 1 c + k 1, c 0 1 stands for the Pochhammer symbol; note that the above series terminates if one of the a i is a negative integer. Solving the eigenvalue equation it is found that the eigenvalues λ n are given by [7] K 1 B n x = λ n B n x, n = 0, 1,,... 7 λ n = 1 n n + h, 8 3

and that the polynomials have the expression 4F 3 B n x c n = where the coefficient n, n+1 +h, x r 1+1/, x r 1 +1/ 1 r 1 r, ρ 1 r 1 + 1, ρ r 1 + 1 + n x r 1+ 1 ρ 1 r 1 + 1 ρ r 1 + 1 4 F 3 4F 3 n 1, n +h, x r 1+ 1, x r 1+ 1 1 r 1 r, ρ 1 r 1 + 1, ρ r 1 + 1 1 1 n, n+1 +h, x r 1+3/, x r 1 +1/ 1 r 1 r, ρ 1 r 1 + 3, ρ r 1 + 3 1 n +hx r 1+ 1 ρ 1 r 1 + 1 ρ r 1 + 1 4 F n 1, n+ +h, x r 1+ 3, x r 1+ 1 3 1 r 1 r, ρ 1 r 1 + 3, ρ r 1 + 3 1 1 n even, n odd, 9 c n+p = 1 p 1 r 1 r n ρ 1 r 1 + 1/, ρ r 1 + 1/ n+p n + h + 1/ n+p, p {0, 1}, ensures that the polynomials B n x are monic, i.e. B n x = x n + Ox n 1. The polynomials 9 were first written down by Bannai and Ito in their classification of the orthogonal polynomials satisfying 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] has allowed to develop their theory. That they obey a three-term difference equation stems from the fact that there are grids such as for which operators of the form x s = 1 s s/ + a + 1/4 1/4, H = AxR + BxT + R + Cx, are tridiagonal in the basis fx s { Bx s fx s+1 + Ax s fx s 1 + Cx s fx s Hfx s = Ax s fx s+1 + Bx s fx s 1 + Cx s fx s s even, s odd. It was observed by Bannai and Ito that the polynomials 9 correspond to a q 1 limit of the q-racah polynomials see [11] for the definition of q-racah polynomials. In this connection, it is worth mentioning that the Bannai-Ito algebra 5 generated by the defining operator K 1 and the recurrence operator K of the Bannai-Ito polynomials can be obtained as a q 1 limit of the Zhedanov algebra [1], which encodes the bispectral property of the q-racah polynomials. The Bannai-Ito polynomials B n x have companions I n x = B n+1x B n+1ρ 1 B nρ 1 B nx x ρ 1, called the complementary Bannai-Ito polynomials [13]. It has now been understood that the polynomials B n x and I n x are the ancestors of a rich ensemble of polynomials referred to as 1 orthogonal polynomials [7, 13, 14, 15, 16, 17, 18]. All polynomials of this scheme are eigenfunctions of first or second order operators of Dunkl type, i.e. which involve reflections. 4

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 relation satisfied by the Bannai-Ito polynomials. In order to obtain this relation, one needs to find the action of the operator K on the BI polynomials B n x. Introduce the operators K + = K + K 3 K 1 1/ ω + ω 3, K = K K 3 K 1 + 1/ + ω ω 3, 10 where K i and ω i are given by, 3, 4 and 6. It is readily checked using 5 that { K 1, K ± } = ±K ±. One can directly verify that K ± maps polynomials to polynomials. In view of the above, one has K 1 K+ B n x = K + K1 + K + B n x = 1 λ n K + B n x, where λ n is given by 8. It is also seen from 8 that { λ n 1 n even, 1 λ n = n odd. λ n+1 It follows that Similarly, one finds K + B n x = K B n x = { α n 0 B n 1 x n even, α n 1 B n+1 x n odd. { β n 0 B n+1 x n even, β n 1 B n 1 x n odd. The coefficients α 0 n = n n + ρ 1 + ρ r 1 + r n n 1 n + h 1 + h, α 1 n = 4n + h + 1/, β n 0 = 4n + h + 1/, β n 1 = 4ρ 1 r 1 + n ρ r 1 + n ρ 1 r + n ρ r + n, n + h 1/ can be obtained from the comparison of the highest order term. Introduce the operator From the definition 10 of K ±, it follows that V = K + K 1 + 1/ + K K 1 1/. 11 V = K K 1 1/4 ω 3 K1 ω /. 1 From 7, 11 and the actions of the operators K ±, we find that V is two-diagonal { λ n + 1/α n 0 B n 1 x + λ n 1/β n 0 B n+1 x n even, V B n x = λ n 1/β n 1 B n 1 x + λ n + 1/α n 1 B n+1 x n odd. 13 5

From 1 and recalling the definition 3 of K, we have also V B n x = [ λ n 1/44x + 1 ω 3 λ n ω / ] B n x. 14 Upon combining 13 and 14, one finds that the Bannai-Ito polynomials satisfy the three-term recurrence relation where x B n x = B n+1 x + ρ 1 A n C n B n x + A n 1 C n B n 1 x, A n = C n = { n+1+ρ1 r 1 n+1+ρ 1 r 4n+ρ 1 +ρ r 1 r +1 n+1+ρ 1 +ρ r 1 r n+1+ρ 1 +ρ 4n+ρ 1 +ρ r 1 r +1 { nn r 1 r 4n+ρ 1 +ρ r 1 r n+ρ r n+ρ r 1 4n+ρ 1 +ρ r 1 r n even, n odd. n even, n odd, 15 The positivity of the coefficient A n 1 C n restricts the polynomials B n x to being orthogonal on a finite set of points [19]. 6. The paraboson algebra and sl 1 The next realization of the Bannai-Ito algebra will involve sl 1 ; this algebra, introduced in [0], is closely related to the parabosonic oscillator. 6.1. The paraboson algebra Let a and a be the generators of the paraboson algebra. These generators satisfy [1] [{a, a }, a] = a, [{a, a }, a ] = a. Setting H = 1 {a, a }, the above relations amount to [H, a] = a, [H, a ] = a, which correspond to the quantum mechanical equations of an oscillator. 6.. Relation with osp1 The paraboson algebra is related to the Lie superalgebra osp1 []. Indeed, upon setting F = a, F + = a, E 0 = H = 1 {F +, F }, E + = 1 F +, E = 1 F, and interpreting F ± as odd generators, it is directly verified that the generators F ±, E ± and E 0 satisfy the defining relations of osp1 [3]: [E 0, F ± ] = ±F ±, {F +, F } = E 0, [E 0, E ± ] = ±E ±, [E, E + ] = E 0, The osp1 Casimir operator reads [F ±, E ± ] = 0, [F ±, E ] = F. C osp1 = E 0 1/ 4E + E F + F. 6

6.3. sl q Consider now the quantum algebra sl q. It can be presented in terms of the generators A 0 and A ± satisfying the commutation relations [4] Upon setting these relations become The sl q Casimir operator is of the form [A 0, A ± ] = ±A ±, [A, A + ] = qa 0 q A 0 q q 1. B + = A + q A 0 1/, B = q A 0 1/ A, B 0 = A 0, [B 0, B ± ] = ±B ±, B B + qb + B = qb 0 1 q 1. C slq = B + B q B 0 q 1q 1 qb 0 1 + q B 0. Let j be a non-negative integer. The algebra sl q admits a discrete series representation on the basis j, n with the actions q B 0 j, n = q j+n j, n, n = 0, 1,,.... The algebra has a non-trivial coproduct : sl q sl q sl q which reads B 0 = B 0 1 + 1 B 0, B ± = B ± q B 0 + 1 B ±. 6.4. The sl 1 algebra as a q 1 limit of sl q The sl 1 algebra can be obtained as a q 1 limit of sl q. Let us first introduce the operator R defined as It is easily seen that R = lim q 1 qb 0. R j, n = 1 j+n j, n = ɛ 1 n j, n, where ɛ = ±1 depending on the parity of j, thus R = 1. When q 1, one finds that q B 0 B + = qb + q B 0 B q B 0 = qq B 0 B {R, B ± } = 0, B B + qb + B = qb 0 1 q 1 {B +, B } = B 0, C slq B + B R B 0 R + R/, B ± = B ± q B 0 + 1 B ± B ± = B ± R + 1 B ±. In summary, sl 1 is the algebra generated by J 0, J ± and R with the relations [0] [J 0, J ± ] = ±J ±, [J 0, R] = 0, {J ±, R} = 0, {J +, J } = J 0, R = 1. 16 7

The Casimir operator has the expression and the coproduct is of the form [5] Q = J + J R J 0 R + R/, 17 J 0 = J 0 1 + 1 J 0, J ± = J ± R + 1 J ±, R = R R. 18 The sl 1 algebra 16 has irreducible and unitary discrete series representations with basis ɛ, µ; n, where n is a non-negative integer, ɛ = ±1 and µ is a real number such that µ > 1/. These representations are defined by the following actions: J 0 ɛ, µ; n = n + µ + 1 ɛ, µ; n, R ɛ, µ; n = ɛ 1n ɛ, µ; n, J + ɛ, µ; n = ρ n+1 ɛ, µ; n + 1, J ɛ, µ; n = ρ n ɛ, µ; n 1, where ρ n = n + µ1 1 n. In these representations, the Casimir operator takes the value Q ɛ, µ; n = ɛµ ɛ, µ; n. These modules will be denoted by V ɛ,µ. Let us offer the following remarks. The sl 1 algebra corresponds to the parabose algebra supplemented by R. The sl 1 algebra consists of the Cartan generator J 0 and the two odd elements of osp1 supplemented by the involution R. One has C osp1 = Q, where Q is given by 17. Thus the introduction of R allows to take the square-root of C osp1. In sl 1, one has [J, J + ] = 1 QR. On the module V ɛ,µ, this leads to [J, J + ] = 1 + ɛµr. 7. Dunkl operators The irreducible modules V ɛ,µ of sl 1 can be realized by Dunkl operators on the real line. Let R x be the reflection operator The Z -Dunkl operator on R is defined by [6] R x fx = f x. D x = x + ν x 1 R x, where ν is a real number such that ν > 1/. Upon introducing the operators Ĵ ± = 1 x D x, and defining Ĵ0 = 1 {Ĵ, Ĵ+}, it is readily verified that a realization of the sl 1 -module V ɛ,µ with ɛ = 1 and µ = ν is obtained. In particular, one has [Ĵ, Ĵ+] = 1 + νr x. It can be seen that Ĵ ± = Ĵ with respect to the measure x ν dx on the real line [7]. 8

8. The Racah problem for sl 1 and the Bannai-Ito algebra The Racah problem for sl 1 presents itself when the direct product of three irreducible representations is examined. We consider the three-fold tensor product V = V ɛ 1,µ 1 V ɛ,µ V ɛ 3,µ 3. It follows from the coproduct formula 18 that the generators of sl 1 on V are of the form J 4 = J 1 0 + J 0 + J 3 0, J 4 ± = J 1 ± R R 3 + J ± R3 + J 3 ±, R4 = R 1 R 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 to bring 3 after or one can combine and 3 before adding 1. This is represented by V ɛ 1,µ 1 V ɛ,µ V ɛ 3,µ 3 or V ɛ 1,µ 1 V ɛ,µ V ɛ 3,µ 3. 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 Ri J i 0 1/R i = ɛ i µ i, i = 1,, 3. The intermediate Casimir operators Q ij = J i + Rj + J j i + J Rj + J j Ri R j J i 0 + J j 0 1/R i R j = J i J j + J i + J j Ri R i R j / + Q i R j + Q j R i, where ij = 1, 3. The total Casimir operator Q 4 = [J 4 + J 4 4 J 0 1/]R 4. Let q 1, q 4 ; m and q 3, q 4 ; m be the orthonormal bases associated to the two coupling schemes presented in 19. These two bases are defined by the relations and Q 1 q 1, q 4 ; m = q 1 q 1, q 4 ; m, Q 3 q 3, q 4 ; m = q 3 q 3, q 4 ; m, Q 4, q 4 ; m = q 4, q 4 ; m, J 4 0, q 4; m = m + µ 1 + µ + µ 3 + 3/, q 4 ; m. The Racah problem consists in finding the overlap coefficients q 3, q 4 q 1, q 4, between the eigenbases of Q 1 and Q 3 with a fixed value q 4 of the total Casimir operator Q 4 ; as these coefficients do not depend on m, we drop this label. For simplicity, let us now take ɛ 1 = ɛ = ɛ 3 = 1. 9

Upon defining K 1 = Q 3, K 3 = Q 1, one finds that the intermediate Casimir operators of sl 1 realize the Bannai-Ito algebra [8] with structure constants {K 1, K 3 } = K + Ω, {K 1, K } = K 3 + Ω 3, {K, K 3 } = K 1 + Ω 1, 0 Ω 1 = µ 1 µ + µ µ 3, Ω = µ 1 µ 3 + µ µ, Ω 3 = µ 1 µ + µ 3 µ, 1 where µ = ɛ 4 µ 4 = q 4. The first relation in 0 can be taken to define K which reads K = J 1 + J 3 J 1 J 3 + R1 R + R 1 R 3 / Q 1 R 3 Q 3 R 1. In the present realization the Casimir operator of the Bannai-Ito algebra becomes Q BI = µ 1 + µ + µ 3 + µ 4 1/4. It has been shown in section 3 that the Bannai-Ito polynomials form a basis for a representation of the BI algebra. It is here relatively easy to construct the representation of the BI algebra on bases of the three-fold tensor product module V with basis vectors defined as eigenvectors of Q 1 or of Q 3. The first step is to obtain the spectra of the intermediate Casimir operators. Simple considerations based on the nature of the sl 1 representation show that the eigenvalues q 1 and q 3 of Q 1 and Q 3 take the form [9, 30, 8, 0]: q 1 = 1 s 1+1 s 1 + µ 1 + µ + 1/, q 3 = 1 s 3 s 3 + µ + µ 3 + 1/, where s 1, s 3 = 0, 1,..., N. The non-negative integer N is specified by N + 1 = µ 4 µ 1 µ µ 3. Denote the eigenstates of K 3 by k and those of K 1 by s ; one has K 3 k = 1 k k + µ 1 + µ + 1/ k, K 1 s = 1 s s + µ + µ 3 + 1/ s. Given the expressions 1 for the structure constants Ω k, one can proceed to determine the N + 1 N + 1 matrices that verify the anticommutation relations 0. The action of K 1 on k is found to be [8]: K 1 k = U k+1 k + 1 + V k k + U k k 1, with V k = µ + µ 3 + 1/ B k D k and U k = B k 1 D k where B k = D k = { k+µ +1k+µ 1 +µ +µ 3 µ+1 k+µ 1 +µ +1 k+µ 1 +µ +1k+µ 1 +µ +µ 3 +µ+1 k+µ 1 +µ +1 { kk+µ 1 +µ µ 3 µ k+µ 1 +µ k+µ 1k+µ 1 +µ µ 3 +µ k+µ 1 +µ n even, n odd. k even, k odd, 10

Under the identifications ρ 1 = 1 µ + µ 3, ρ = 1 µ 1 + µ, r 1 = 1 µ 3 µ, r = 1 µ µ 1, 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 one has on the one hand and on the other hand Comparing the two RHS yields s k = ws k B k x s, B 0 x s 1, s K 1 k = 1 s s + ρ 1 + 1/ s k, s K 1 k = U k+1 s k + 1 + V k s k + U k 1 s k 1. x s B k x s = B k+1 x s + ρ 1 A k C k B k x s + A k 1 C k B k 1 x s, where x s are the points of the Bannai-Ito grid x s = 1 s s + ρ 1 + 1/4 1/4, s = 0,..., N. Hence the Racah coefficients of sl 1 are proportional to the Bannai-Ito polynomials. The algebra 0 with structure constants 1 is invariant under the cyclic permutations of the pairs K i, µ i. As a result, the representations in the basis where K 1 is diagonal can be obtained directly. In this basis, the operator K 3 is seen to be tridiagonal, which proves again that the Bannai-Ito polynomials possess the Leonard duality property. 9. A superintegrable model on S with Bannai-Ito symmetry We shall now use the analysis of the Racah problem for sl 1 and its realization in terms of Dunkl operators to obtain a superintegrable model on the two-sphere. Recall that a quantum system in n dimensions with Hamiltonian H is maximally superintegrable it it possesses n 1 algebraically independent constants of motion, where one of these constants is H [31]. Let s 1, s, s 3 R and take s 1 + s + s 3 = 1. The standard angular momentum operators are L 1 = 1 i s s 3, L = 1 s 3 s i The system governed by the Hamiltonian H = L 1 + L + L 3 + µ 1 s 1 s 3 s 1, L 3 = 1 s 1 s 3 i µ 1 R 1 + µ s s 1 s. s s 1 µ R + µ 3 s µ 3 R 3, 3 with µ i, i = 1,, 3, real parameters such that µ i > 1/ is superintegrable [3]. i The operators R i reflect the variable s i : R i fs i = f s i. ii The operators R i commute with the Hamiltonian: [H, R i ] = 0. 11

iii If one is concerned with the presence of reflection operators in a Hamiltonian, one may replace R i by κ i = ±1. This then treats the 8 potential terms µ 1 s 1 µ 1 κ 1 + µ s µ κ + µ 3 s µ 3 κ 3, 3 simultaneously much like supersymmetric partners. iv Rescaling s i rs i and taking the limit as r gives the Hamiltonian of the Dunkl oscillator [7, 33] H = [D x 1 + D x ] + µ 3x 1 + x, after appropriate renormalization; see also [34, 35, 36]. It can be checked that the following three quantities commute with the Hamiltonian [9, 3]: s 3 s C 1 = il 1 + µ R µ 3 R 3 R + µ R 3 + µ 3 R + R R 3 /, s s 3 s 3 s 1 C = il + µ 1 R 1 µ 3 R 3 R 1 R + µ 1 R 3 + µ 3 R 1 + R 1 R 3 /, s 1 s 3 s s 1 C 3 = il 3 + µ 1 R 1 µ R R 1 + µ 1 R + µ R 1 + R 1 R /, s 1 s that is, [H, C i ] = 0 for i = 1,, 3. To determine the symmetry algebra generated by the above constants of motion, let us return to the Racah problem for sl 1. Consider the following gauge transformed parabosonic realization of sl 1 in the three variables s i : J i ± = 1 [ s i si ± µ i R i s i ], J i 0 = 1, [ s i + s i + µ ] i s µ i R i, R i = R i, 3 i for i = 1,, 3. Consider also the addition of these three realizations so that J 0 = J 1 0 + J 0 + J 3 0, J ± = J 1 ± R R 3 + J ± R3 + J 3 ±, R = R1 R R 3. 4 It is observed that in the realization 4, the total Casimir operator can be expressed in terms of the constants of motion as follows: Upon taking one finds Q = C 1 R 1 C R C 3 R 3 + µ 1 R R 3 + µ R 1 R 3 + µ 3 R 1 R + R/. Ω + Ω = L 1 + L + L 3 + s 1 + s + s 3 Ω = QR, µ1 s 1 µ 1 R 1 + µ s µ R + µ 3 s µ 3 R 3, 5 3 so that H = Ω +Ω if s 1 +s +s 3 = 1. Assuming this constraint can be imposed, H is a quadratic combination of QR. By construction, the intermediate Casimir operators Q ij commute with the total Casimir operator Q and with R and hence with Ω; they thus commute with H = Ω + Ω and are the constants of motion. It is indeed found that Q 1 = C 3, Q 3 = C 1, 1

in the parabosonic realization 3. Let us return to the constraint s 1 + s + s 3 = 1. Observe that 1 J + + J = s 1 R R 3 + s R 3 + s 3 = s 1 + s + s 3. Because J + + J commutes with Ω = QR, Q 1 and Q 3, one can impose s 1 + s + s 3 = 1. Since it is already known that the intermediate Casimir operators in the addition of three sl 1 representations satisfy the Bannai-Ito structure relations, the constants of motion verify {C 1, C } = C 3 µ 3 Q + µ 1 µ, {C, C 3 } = C 1 µ 1 Q + µ µ 3, {C 3, C 1 } = C µ Q + µ 3 µ 1, and thus the symmetry algebra of the superintegrable system with Hamiltonian is a central extension with Q begin the central operator of the Bannai-Ito algebra. Let us note that the relation H = Ω + Ω relates to chiral supersymmetry since with S = Ω + 1/ one has 10. A Dunkl-Dirac equation on S Consider the Z -Dunkl operators D i = 1 {S, S} = H + 1/4. x i + µ i x i 1 R i, i = 1,,..., n, with µ i > 1/. The Z n -Dunkl-Laplace operator is D = n Di. i=1 With γ n the generators of the Euclidean Clifford algebra the Dunkl-Dirac operator is {γ m, γ n } = δ nm, /D = n γ i D i. i=1 Clearly, one has /D = D. Let us consider the three-dimensional case. Introduce the Dunkl angular momentum operators J 1 = 1 i x D 3 x 3 D, J = 1 i x 3D 1 x 1 D 3, J 3 = 1 i x 1D x D 1. Their commutation relations are found to be [J i, J k ] = iɛ jkl J l 1 + µ l R l. 6 13

The Dunkl-Laplace equation separates in spherical coordinates; i.e. one can write D = D 1 + D + D 3 = M r + 1 r S, where S is the Dunkl-Laplacian on the -sphere. It can be verified that [36] J = J 1 + J + J 3 = S + µ 1 µ 1 R 1 R + µ µ 3 1 R R 3 + µ 1 µ 3 1 R 1 R 3 µ 1 R 1 µ R µ 3 R 3 + µ 1 + µ + µ 3. In three dimensions the Euclidean Clifford algebra is realized by the Pauli matrices 0 1 0 i 1 0 σ 1 =, σ 1 0 =, σ i 0 3 =, 0 1 which satisfy Consider the following operator: σ i σ j = iɛ ijk σ k + δ ij. Γ = σ J + µ R, 7 with µ R = µ 1 R 1 + µ R + µ 3 R 3. Using the commutation relations 6 and the expression 7 for J, it follows that Γ + Γ = S + µ 1 + µ + µ 3 µ 1 + µ + µ 3 + 1. This is reminiscent of the expression 5 for the superintegrable system with Hamiltonian in terms of the sl 1 Casimir operator. This justifies calling Γ a Dunkl-Dirac operator on S since a quadratic expression in Γ gives S. The symmetries of Γ can be constructed. They are found to have the expression [37] M i = J i + σ i µ j R j + µ k R k + 1/, and one has [Γ, M i ] = 0. It is seen that the operators also commute with Γ. Furthermore, one has X i = σ i R i i = 1,, 3 ijk cyclic, [M i, X i ] = 0, {M i, X j } = {M i, X k } = 0. Note that Y = ix 1 X X 3 = R 1 R R 3 is central like Γ. The commutation relations satisfied by the operators M i are [M i, M j ] = iɛ ijk M k + µ k Γ + 1X k + µ i µ j [X i, X j ]. This is again an extension of su 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 1, K } = K 3 + µ 3 Γ + 1Y + µ 1 µ, {K, K 3 } = K 1 + µ 1 Γ + 1Y + µ µ 3, {K 3, K 1 } = K + µ 3 Γ + 1Y + µ 3 µ 1, 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. 14

11. Conclusion In this paper, we have presented the Bannai-Ito algebra together with some of its applications. In concluding this overview, we identify some open questions. i Representation theory of the Bannai-Ito algebra Finite-dimensional representations of the Bannai-Ito algebra associated to certain models were presented. However, the complete characterization of all representations of the Bannai- Ito algebra is not known. ii Supersymmetry The parallel with supersymmetry has been underscored at various points. One may wonder if there is a deeper connection. iii Dimensional reduction It is well known that quantum superintegrable models can be obtained by dimensional reduction. It would be of interest to adapt this framework in the presence of reflections operators. Could the BI algebra can be interpreted as a W -algebra? iv Higher ranks Of great interest is the extension of the Bannai-Ito algebra to higher ranks, in particular for many-body applications. In this connection, it can be expected that the symmetry analysis of higher dimensional superintegrable models or Dunkl-Dirac equations will be revealing. Acknowledgments V.X.G. holds an Alexander-Graham-Bell fellowship from the Natural Science and Engineering Research 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. References [1] Arik M and Kayserilioglu U 003 Int. J. Mod. Phys. A 18 5039 5046 [] Gorodnii M and Podkolzin G 1984 Irreducible representations of a graded Lie algebra Inst. Math. Acad. Sci. pp 66 76 Spectral theory of operators and infinite-dimensional analysis [3] Brown G 013 Elec. J. Lin. Alg. 6 58 99 [4] Ostrovskii V and Sil vestrov S 199 Ukr. Math. J. 44 1395 1401 [5] Jafarov E I, Stoilova N I and Van der Jeugt J 011 J. Phys. A: Math. Theor. 44 6503 [6] Jafarov E I, Stoilova N I and Van der Jeugt J 011 J. Phys. A: Math. Theor. 44 35505 [7] Tsujimoto S, Vinet L and Zhedanov A 01 Adv. Math. 9 13 158 [8] Andrews G, Askey R and Roy R 1999 Special functions Cambridge: Cambridge University Press [9] Leonard D 198 SIAM J. Math. Anal. 13 656 663 [10] Bannai E and Ito T 1984 Algebraic Combinatorics I: Association Schemes San Francisco: Benjamin/Cummings [11] Koekoek R, Lesky P and Swarttouw R 010 Hypergeometric orthogonal polynomials and their q-analogues 1st ed Springer ISBN 978-3-64-05013-8 [1] Zhedanov A 1991 Theor. Math. Phys. 89 1146 1157 [13] Genest V X, Vinet L and Zhedanov A 013 SIGMA 9 18 37 [14] Genest V X, Vinet L and Zhedanov A 014 SIGMA 10 38 55 [15] Vinet L and Zhedanov A 011 J. Phys. A: Math. Theor. 44 08501 [16] Vinet L and Zhedanov A 01 Trans. Amer. Math. Soc. 364 5491 5507 [17] Tsujimoto S, Vinet L and Zhedanov A 013 Proc. Amer. Math. Soc. 141 959 970 [18] Tsujimoto S, Vinet L and Zhedanov A 011 J. Math. Phys. 5 10351 [19] Chihara T 011 An Introduction to Orthogonal Polynomials Dover Publications [0] Tsujimoto S, Vinet L and Zhedanov A 011 SIGMA 7 93 105 [1] Green H S 1953 Phys. Rev. 90 70 73 15

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