Embeddings of the Racah Algebra into the Bannai-Ito Algebra

Embeddings of the Racah algebra into the Bannai-Ito algebra are proposed in two realizations. First, quadratic combinations of the Bannai-Ito algebra generators in their standard realization on the space of polynomials are seen to generate a central extension of the Racah algebra. The result is also seen to hold independently of the realization. Second, the relationship between the realizations of the Bannai-Ito and Racah algebras by the intermediate Casimir operators of the $\mathfrak{osp}(1|2)$ and $\mathfrak{su}(1,1)$ Racah problems is established. Equivalently, this gives an embedding of the invariance algebra of the generic superintegrable system on the two-sphere into the invariance algebra of its extension with reflections, which are respectively isomorphic to the Racah and Bannai-Ito algebras.


Introduction
The purpose of this paper is to examine the relationship between two algebraic structures that have received some attention in the literature as of late: the Racah algebra and the Bannai-Ito algebra. Using two realizations of these algebras, it will be shown that the former can be embedded into the latter.
The Racah algebra can be described as the algebraic structure encoding the bispectrality of the Racah and Wilson polynomials, which sit at the top the Askey scheme of hypergeometric orthogonal polynomials [5]. This assertion originates from the fact that the Racah algebra (1) has realizations on the space of polynomials, called standard realizations, in which the generators are the difference and recurrence operators associated to these polynomials. As an example, in the case of the Racah polynomials R n (λ(x); α, β, γ; N ) (in the notation of [13]) of degree n in the variable λ(x) = x(x + γ + δ + 1), the standard realization is defined as where T ± f (x) = f (x ± 1) are discrete shift operators and where B(x) and D(x) are of the form The operators (3) satisfy the relations (1) with the structure constants expressed in terms of the parameters α, β, γ and N [5]. In the realization (3), the Casimir operator (2) is a multiple of the identity. In the basis {v i = R i (λ(x); α, β, γ; N )} N i=0 , the realization (3) yields a 3-parameter family of (N + 1)-dimensional irreducible representations of (1) in which κ 1 is represented by a diagonal matrix and κ 2 is represented by a tridiagonal matrix. Infinite-dimensional representations of (1) are obtained by taking the standard representation associated with the difference and recurrence operators of the Wilson polynomials.
The Racah algebra has a an alternative presentation in terms of generators A, B, C and ∆. In this presentation, one has where Γ is a constant or a central operator. The commutation relations are where i A , i B , i C are structure constants. The details of the passage from (1) to (4) can be found in [6]. The presentation (4) is known as the democratic or equitable presentation, as it is invariant under cyclic permutations of the generators A, B and C. As far as we know, the democratic presentation of the Racah algebra first appeared in the work of Lévy-Leblond and Lévy-Nahas (equations (2.8) and (2.9) of [15]), who were concerned with the symmetrical coupling of three angular momenta. The algebra (4), which has also been studied in [1], can be seen as a q = 1 analog of the Askey-Wilson algebra in its Z 3 -symmetric presentation, see [17]. The Bannai-Ito algebra was first introduced in 2012 by Tsujimoto, Vinet and Zhedanov in their study of the Bannai-Ito polynomials [19]. This algebra can be defined as the unital associative algebra over C with generators X, Y , Z and relations where ω X , ω Y , ω Z are structure constants. The Bannai-Ito algebra has a Casimir operator which commutes with all generators. Again, only two of the generators are genuinely independent. The Bannai-Ito algebra encodes the bispectral properties of the Bannai-Ito polynomials. The standard realization of the Bannai-Ito algebra on the space of polynomials is defined as where Rf (z) = f (−z) is the reflection operator, where and where ρ 1 , ρ 2 , r 1 , r 2 are complex parameters. In the realization (7), the Casimir operator (6) acts as a multiple of the identity. If one introduces the basis vectors w i = B i (z; ρ 1 , ρ 2 , r 1 , r 2 ) for i = 0, 1, 2, . . ., where B n (z; ρ 1 , ρ 2 , r 1 , r 2 ) are the Bannai-Ito polynomials of degree n in z (see [2]), one obtains a four-parameter family of infinite-dimensional irreducible representations of (5) where X is represented by a diagonal matrix and Y is represented by a tridiagonal matrix; these representations can be made finite-dimensional under appropriate truncation conditions. The Racah and Bannai-Ito algebras are intimately related to second-order superintegrable systems in two dimensions. Recall that a quantum system in d dimensions governed by a Hamiltonian H is said to be superintegrable if it possesses 2d − 1 algebraically independent constants of motion, see [16] for a review. On the one hand, the Racah algebra arises as the symmetry algebra for the generic superintegrable system on the two-sphere [9,10]. This system is described by the Hamiltonian where k i > −1/2, i = 1, 2, 3, are real parameters and where the operators L i are the standard angular momentum generators in three dimensions The model defined by (8) is the most general second-order superintegrable system in two dimensions (without reflections-see below), as all the other systems in that category can be obtained from it by taking limits and contractions [12]. The connection between the representations of the symmetry algebra generated by the second-order constants of motion of (8) and the Racah/Wilson polynomials was established in [11]. On the other hand, the Bannai-Ito algebra arises as the invariance algebra for the system governed by the Hamiltonian [3] where , are reflection operators. Since the Hamiltonian (9) commutes with the reflection operators, it can be considered as a concatenation of eight systems of type (8), which is reminiscent of supersymmetry. The model described by (9) is in fact supersymmetric, as it can be expressed, up to additive constant, as the square of a Hermitian supercharge (see [3,7]). The fact that the Racah and Bannai-Ito algebras appear as symmetry algebras for the models (8) and (9) is directly related to the fact that these algebras also arise as the commutant algebras for the actions of the Lie algebra su(1, 1) and Lie superalgebra osp(1|2) on three-fold tensor product representations (see [5] and [4]). Indeed, in three-fold tensor product representations involving three irreducible representations of the positive-discrete series for su(1, 1) and osp(1|2), the Hamiltonians (8), (9) and their symmetries respectively correspond to total and intermediate Casimir operators constructed from the coproducts of su(1, 1) and osp(1|2) (see Section 3 for details).
In light of the visible relationship between the two Hamiltonians (8), (9) and the embedding su(1, 1) ⊂ osp(1|2), it is natural to expect a relation between the Racah and Bannai-Ito algebras. In the present paper, this relation will be made explicit.
Here is the outline of the paper. In Section 2, we show that upon taking appropriate quadratic combinations of the generators of the Bannai-Ito algebra (5), one finds a central extension of the Racah algebra in its equitable presentation. In Section 3, the quadratic relation between the sCasimir operator of osp(1|2) and the Casimir operator of su(1, 1) is derived. The realization of the Racah generators in terms of the Bannai-Ito generators in the context of three-fold tensor product representations of su(1, 1) and osp(1|2) is given. The relation between the constants of motion of the superintegrable systems (8) and (9) is deduced. A short conclusion follows.
2 Embedding of the Racah algebra into the Bannai-Ito algebra and the standard realization Consider the standard realization (7) of the Bannai-Ito algebra. In this representation, the structure constants have the expressions and the Casimir operator (6) acts a multiple of the identity with value

Consider the quadratic combination of generators
and introduce the operator It is verified that I(z) commutes the operators A(z), B(z) and C(z), i.e., It is also observed that the generators A(z), B(z), C(z) commute with their sum since one has where U (z) is the Bannai-Ito Casimir operator (6). Let ∆(z) be defined as A direct calculation using the expressions (7) for the standard realization shows that the operators A(z), B(z), C(z) and ∆(z) satisfy the commutation relations Upon comparing the above relations with those appearing in (4), it is directly seen that the above relations coincide with the defining relations of the equitable Racah algebra extended by the central operator I(z). We have thus shown that the quadratic combinations (10) of the generators of the Bannai-Ito algebra (5) in the standard realization (7) realize the Racah algebra. The explicit expression of the generators A(z), B(z) and C(z) should be compared with (3).
Remark 1. While the above calculations have been performed with the help of the faithful model provided by the realization (7), let us observe that the embedding (13) of the Racah algebra into the Bannai-Ito algebra can be obtained directly in an abstract fashion using only the definitions (10), (11), (12), and the relations (5).
3 Embedding of the Racah algebra into the Bannai-Ito algebra, superintegrability in two dimensions and the Racah problem In this section we give another embedding of the Racah algebra into the Bannai-Ito algebra in relation with the realizations of these algebras as symmetry algebras for the superintegrable systems (8) and (9) or equivalently as commutant algebras for the actions of su(1, 1) and osp(1|2) on three-fold tensor product representations. Let us first review how the Racah and Bannai-Ito algebras arise in the context of the latter.

The Racah algebra and the Lie algebra su(1, 1)
The su(1, 1) algebra has three generators K 0 , K ± satisfying the commutation relations The Casimir operator for su(1, 1) can be written as As a Lie algebra, su(1, 1) admits a coproduct ∆ : su(1, 1) → su(1, 1) ⊗ su(1, 1) given by The coproduct allows to construct tensor product representations. Let V (λ) be an irreducible representation space for su(1, 1) on which the Casimir operator C acts as a λI. Consider the three-fold tensor product representation on . The action of the generators is given by The elements that commute with the action of su(1, 1) on V are easily constructed. First, one has the three initial Casimir operators C (i) , C (2) , C (3) for i = 1, 2, 3 associated to each factor V (λ i ) of the tensor product. These operators are defined as Second, one has the total Casimir operator C (4) that has the expression and which acts non-trivially on all factors of the tensor product. Finally, one has the two intermediate Casimir operators C (12) and C (23) that are of the form and which act non-trivially only on a pair of representation spaces. By construction, the initial, intermediate and total Casimir operators commute with action (16) of the su(1, 1) generators on the tensor product space V . However, these Casimir operators do not all commute with each other and they thus generate a non commutative commutant algebra. This algebra can be identified with the Racah algebra as follows.
Let A and C be defined as A = C (23) , C = C (12) , and let B = we thus have Since A, B and C all commute with the Casimir operators C (i) for i = 1, 2, 3, 4, one can introduce the operator Ω defined as Using the expressions of the intermediate and total Casimir operators computed from (14) and the coproduct (15), an involved calculation shows that the operators A, B, C together with the operator Ω, satisfy the commutation relations where the substitutions C (i) = λ i for i = 1, 2, 3, 4 were made to underscore the fact that each eigenspace of C (4) supports a representation of the Racah algebra (17). The relationship between the results presented above and the analysis of the generic superintegrable system on the 2-sphere is as follows [5]. Let V (k) be the representation associated with the following realization of su(1, 1): where k is a real number such that k > −1/2. In this realization, one has This realization corresponds to positive-discrete series [20]. Consider the tensor product representation V (k 1 ) ⊗ V (k 2 ) ⊗ V (k 3 ) obtained by taking three copies of the realization (18) with parameters k 1 , k 2 and k 3 in the three independent variables x 1 , x 2 and x 3 . In this realization, the total Casimir operator C (4) takes the form Upon comparing with (8), it is directly seen that the total Casimir operator corresponds, up to an inessential affine transformation, to the Hamiltonian for the generic superintegrable system on the two-sphere. By construction, the intermediate Casimir operators A, B and C commute with C (4) , and are thus the constants of motion of the Hamiltonian. In this realization, one has The constants of motion A(x 1 , x 2 , x 3 ), B(x 1 , x 2 , x 3 ) and C(x 1 , x 2 , x 3 ) satisfy the defining relations of the equitable Racah algebra (17)

The Bannai-Ito algebra and the Lie superalgebra osp(1|2)
The osp(1|2) Lie superalgebra has two odd generators A ± and one even generator A 0 that obey the commutation relations The even subalgebra of osp(1|2) is generated by the elements J ± = A 2 ± and A 0 . The generators of the even subalgebra satisfy the defining relations of su(1, 1). One has indeed whence it follows that su(1, 1) ⊂ osp(1|2). The abstract Z 2 grading of osp(1|2) can be concretized by appending the grade involution P to the set of generators and declaring that the even generators commute with P and that the odd generators anticommute with P . Hence one can define osp(1|2) as the algebra generated by A ± , A 0 and the involution P satisfying the relations This presentation has been referred to as sl −1 (2) [18]; in this presentation the grading of the elements no longer needs to be specified. The sCasimir operator S of osp(1|2) is defined as [14] The sCasimir operator satisfies the relations There is a quadratic relation between the sCasimir of osp(1|2) and the Casimir operator of the even su(1, 1) subalgebra, denoted by C su(1,1)⊂osp(1|2) . A direct calculation shows that where one has as per (14). An operator that commutes with every element of osp(1|2) can be constructed by combining the sCasimir with the grade involution. It is indeed verified that the operator commutes with A ± and A 0 . We shall refer to (22) as the Casimir operator for osp(1|2).
The osp(1|2) algebra has a coproduct defined as follows This coproduct allows to construct tensor product representations. Let W (λ) be an irreducible representation space for osp(1|2) on which the Casimir operator (22) acts as λI. Consider the three-fold tensor product representation W = W (λ 1 ) ⊗ W (λ 2 ) ⊗ W (λ 3 ) . The action of the osp(1|2) generators is Once again, the elements that commute with this action are easily found. First, one has the initial Casimir operators Q (i) , Q (2) , Q (3) for i = 1, 2, 3 associated to each factor W (λ i ) . They are defined as Second, one has the total Casimir operator Q (4) defined as which acts non-trivially on all factors of the three-fold tensor product. Finally, one has the two intermediate Casimir operators Q (12) and Q (23) given by which act non-trivially only on a pair of representation spaces. By construction, the initial, intermediate and total Casimir operators commute with the action of the osp(1|2) generators on the tensor product space W . However, these Casimir operators do not all commute with each other and they thus give rise to a non-commutative commutant algebra. This algebra can be identified with the Bannai-Ito algebra in the following way. Let Z and X be defined by Upon using the formulas (22) and (23) for the Casimir and coproduct for osp(1|2), an involved calculation shows that the operators Z, X satisfy the relations where Y is defined by the first relation of (24) and where Again we have made the substitutions Q (i) → −λ i for i = 1, 2, 3, 4, replacing the initial and total Casimir operators, which commute with X, Y , Z, by constants. The relationship between the superintegrable system with reflections on the two-sphere governed by the Hamiltonian (9) and the above considerations can be established as follows. Let W (µ) be the representation of osp(1|2) associated with the coordinate realization where R x f (x) = f (−x) is the reflection operator and where µ is a real parameter such that µ > 0. In this realization, the Casimir operator Q of osp(1|2) acts as a multiple of the identity with multiple The Casimir operator for the even su(1, 1) subalgebra has the expression Consider now the tensor product representation W = W (µ 1 ) ⊗ W (µ 2 ) ⊗ W (µ 3 ) obtained by combining three copies of the realization (26) in three independent variables x 1 , x 2 , x 3 using the coproduct specified in (23). In this realization one has where S (4) is the total sCasimir operator. It is directly seen that the right-hand side of (3.2) corresponds to the Hamiltonian for the generic superintegrable system on the 2-sphere with reflections. The supersymmetry of (3.2) is manifest from the above equation. Indeed, the right-hand side of (3.2) can be written, up to a constant, as the square of the Hermitian supercharge S (4) . The intermediate Casimir operators X and Z as well as the derived operator Y commute with the total Casimir Q (4) , but also with the total sCasimir S (4) . Indeed, in view of (22) and (23), one has Since the operators X, Y , Z commute with the product R 1 R 2 R 3 , it follows that they are symmetries of S (4) and consequently of the Hamiltonian (9). In the present realization, the constants of motion have the expressions They satisfy the commutations relations (24) with the structure constants (25). In this realization, the Casimir operator of the Bannai-Ito algebra has the expression

Relationship between the two realizations
Let us now make explicit the relationship between the realization (20) of the Racah algebra (17) by the constants of motion of the generic superintegrable system on the 2-sphere governed by the Hamiltonian (19) and the realization (28) of the Bannai-Ito algebra (24) by the joint symmetries of the total sCasimir operator S (4) and the generic superintegrable system with reflections on the 2-sphere described by the Hamiltonian (3.2). In view of the relation (21), we consider the quadratic combinations Remark 2. Let us note that the combinations (29) differ from those considered in (10) since they involve the reflection operators R 1 , R 2 and R 3 . This is related to the fact that the operators (28) realize in this case a central extension of the Bannai-Ito algebra with the total Casimir operator Q (4) (x 1 , x 2 , x 3 ) arising in the structure "constants", see (25).
A direct calculation shows that the operators A, B and C have the expressions These expressions are seen to correspond to the ones in (20) with k i having the expression It is directly verified that the operators A(x 1 , x 2 , x 3 ), B(x 1 , x 2 , x 3 ) and C(x 1 , x 2 , x 3 ) commute with all the reflections. As a consequence, their algebraic properties are not affected by the operator-valuedness of the k i . It follows that the operators (30) provide a realization of the Racah algebra (17) where for i = 1, 2, 3 and where λ 4 corresponds to the Hamiltonian (19) with the k i as in (31).

Conclusion
In this paper we have provided embeddings of the Racah into the Bannai-Ito algebra in two different realizations. On the one hand, we have shown that in the standard realization of the Bannai-Ito algebra on the space of polynomials, certain quadratic combinations of the Bannai-Ito generators realize a central extension of the Racah algebra. On the other hand, we have shown that in the realization of the Bannai-Ito algebra by the constants of motion of the generic second-order superintegrable on the 2-sphere with reflections, again quadratic combinations of the Bannai-Ito generators involving reflections satisfy the Racah algebra relations with structure constants that depend on reflections.
The results presented here establish that the Racah algebra is embedded into the Bannai-Ito algebra. This supports the assertion that the Bannai-Ito algebra can in fact be considered as more fundamental than the Racah algebra, the former being essentially the "square-root" of the latter. In view of the deep relationship between the Racah and Bannai-Ito algebras with the Racah/Wilson and Bannai-Ito polynomials, it should be possible, in principle, to derive the main properties of the Racah/Wilson polynomials from those of the Bannai-Ito polynomials. It would be of interest to investigate this possibility.