The dual pair $Pin(2n)\times\mathfrak{osp}(1|2)$, the Dirac equation and the Bannai-Ito algebra

The Bannai-Ito algebra can be defined as the centralizer of the coproduct embedding of $\mathfrak{osp}(1|2)$ in $\mathfrak{osp}(1|2)^{\otimes n}$. It will be shown that it is also the commutant of a maximal Abelian subalgebra of $\mathfrak{o}(2n)$ in a spinorial representation and an embedding of the Racah algebra in this commutant will emerge. The connection between the two pictures for the Bannai-Ito algebra will be traced to the Howe duality which is embodied in the $Pin(2n)\times\mathfrak{osp}(1|2)$ symmetry of the massless Dirac equation in $\mathbb{R}^{2n}$. Dimensional reduction to $\mathbb{R}^{n}$ will provide an alternative to the Dirac-Dunkl equation as a model with Bannai-Ito symmetry.


Introduction
The Bannai-Ito algebra B(n) can be presented in terms of generators and relations [1,2]. Let [n] = {1, 2, . . ., n} denote the set of the n first integers and S = {s 1 , . . ., s k } be an ordered k-subset of [n]. The generators Γ S of B(n) are labelled by all subsets for k = 0, 1, . . ., n. For any two subsets A and B of [n], the relations between the generators Γ A and Γ B that define B(n) are: where {X , Y } = X Y + Y X . By convention Γ = − 1 /2, and the generators associated to a set are simply labelled by the indices Γ {i 1 ,...,i k } ≡ Γ i 1 ... i k . Moreover, Γ i , i ∈ [n] and Γ [n] are central.
For the rank one case which occurs when n = 3, the relations for B(3) are seen to be [1,3] {Γ i j , Γ jk } = Γ ik + 2Γ j Γ i jk + 2Γ i Γ k , (1.2) where i, j, k ∈ [3] are all distinct. We shall present in this paper a defining context for B(n) (and B (3)) as a commutant and relate this result to a Howe duality framework.
The Bannai-Ito algebra B(3) was initially introduced in [4] to encode the bispectral properties of the Bannai-Ito polynomials which were discovered by the researchers whose name they bear in a classification problem in algebraic combinatorics [5]. B(3) was later seen [6] to be isomorphic to a degenerate double affine Hecke algebra (DAHA) of type (C ∨ 1 , C 1 ). A key connection between the Lie superalgebra osp(1|2) and the Bannai-Ito algebra was also made [3]. Indeed, the Bannai-Ito polynomials were identified as forming the Racah coefficients for osp(1|2) [7]. It was then shown quite naturally that B(3) is realized by the intermediate Casimir operators arising in the threefold tensor product of osp(1|2) with itself. Extending this construction to the n-fold tensor product osp(1|2) ⊗n led to the definition of B(n) [1]. (More details will be given on this later.) A number of applications for B(n) were subsequently found. For instance, the conserved quantities of a superintegrable model on the (n − 1)-sphere [8,9,10,11] were seen to satisfy the commutation relations (1.1). Of particular relevance to the present study is the fact that B(n) is also the symmetry algebra of the (massless) Dirac-Dunkl [1,12] The Bannai-Ito algebra B(n) has much in common with the Racah algebra R(n) [13]. Like B(n), R(n) can be defined in a tensorial fashion as the algebra formed by the intermediate Casimir operators associated to the n-fold tensor product of the Lie algebra sl(2) with itself. R(n) is also the symmetry algebra of a certain generic superintegrable model on S n−1 (without reflections). The relation between the rank one version R(3) and the Racah polynomials [14] is quite similar to the one between the Bannai-Ito algebra B(3) and the Bannai-Ito polynomials. R(3) has three generators K 1 , K 2 , K 3 which are subjected to the relations with d, e 1 , e 2 central. The defining relations for R(n) [13] are given in Section 7 where an embedding of R(n) into B(n) will be identified.
We have recently observed that R(3) is the commutant in oscillator representations of the universal enveloping algebra U (o(6)) of the subalgebra o(2) ⊕ o(2) ⊕ o(2) of the Lie algebra o(6) of rotations in six dimensions [15]. This was extended to R(n) in [16]. We shall provide here an analogous description of B(n) and shall focus first on B(3).
After recalling in Section 2 how B(3) can be viewed as the centralizer of the coproduct embedding of osp(1|2) into osp(1|2) ⊗3 , we shall show in Section 3 that (2) in the enveloping algebra of the spinorial representation of o(6) associated to the Clifford algebra in R 6 . By considering the (massless) Dirac equation, a Howe duality setting will be brought up to explain the connection between the results of the two previous sections. Dimensional reduction will be performed in Section 5 to complete the picture and will result in a new class of model with Bannai-Ito symmetry. In Section 6 the generalization of these results to the Bannai-Ito algebra B(n) will be obtained for n > 3. As mentioned above, an embedding of the higher rank R(n) Racah algebra in the B(n) Bannai-Ito algebra will be explicitly given in Section 7, thus linking the construction presented here to the one in [16]. Brief concluding remarks will follow. Finally, in Appendix A we indicate that our results offer as a byproduct a derivation through dimensional reduction of the superconformal quantum Hamiltonian of Fubini and Rabinovici which is known to be invariant under sl(2|1) [17].

The superalgebra osp(1|2) and the Bannai-Ito algebra as a centralizer
The superalgebra osp(1|2) can be presented as follows. Let J 0 , J ± be respectively the even and odd generators of the algebra, obeying the relations The Z 2 -grading of the superalgebra can be encoded through a grading involution S , which commutes with even elements and anticommutes with odd elements: The sCasimir operator of osp(1|2) given by commutes with all the odd elements and anticommutes with all the even ones, that It is then straightforward to define a Casimir of osp(1|2) by combining the sCasimir with the involution: This Casimir Γ commutes with all the elements of osp(1|2).
There is a coassociative algebra morphism, the coproduct ∆, which acts as follows: where the superindex denotes on which factor of the tensor product the generator is acting.
Now consider the product of three copies of osp(1|2). The generators corresponding to embeddings in two factors are (2.5) Note the presence of S (2) in J (13) ± . Applying the coproduct twice yields labelled by A ⊂ [3] can then be obtained from the sets above: Even though the intermediate Casimirs Γ A commute with the action of osp(1|2), they do not all commute with each other. A direct computation shows that they precisely obey the commutation relations (1.2) of B(3) thereby proving that this algebra is the centralizer of ∆ (2) (osp(1|2)) in U (osp(1|2) ⊗3 ).

The rank 1 Bannai-Ito algebra as a commutant
We will now show how the Bannai-Ito algebra arises as the commutant of the Let Cℓ 6 be the Clifford algebra generated by the elements γ 1 , . . ., γ 6 verifying the relations Denote by ℓ µν , µ, ν = 1, . . ., 6 the generators of o(6) which obey We shall consider the following representation of the algebra of P in (6) where (3.4) and Σ µν = iγ µ γ ν the spin operators.
We are interested in the commutant of the represented by the set of elements {J 12 , J 34 , J 56 }. First note that: It is observed that the commutant of the span of {J 12 , J 34 , J 56 } is generated by the operators (see the remark below): We can now see that these generate the (rank 1) Bannai-Ito algebra. It is convenient to first introduce the shortened notation j ≡ {2 j − 1, 2 j}, j ∈ N. Take now the elements
Proof: Let us explain how the expression for {Γ 1 (3.10) The additional identity [18] L ab L cd + L ac L db + L ad L bc = 0 (3.11) satisfied in our realization is then the required tool in order to rewrite (3.10) as the r.h.s of (3.8) using the definitions (3.7) and (3.9).

Remark: It is fairly obvious that the elements {G
also belong to the commutant of {J 12 , J 34 , J 56 }. It can be seen that they are algebraically dependent on the Bannai-Ito generators given above. It is interesting to observe however that these G i and K i j realize the Racah algebra R(3) as shown in [15]. This highlights the fact that the Racah algebra can be embedded in the Bannai-Ito algebra.
The explicit embedding and a generalization to the higher rank algebras will be given in Section 7. Operators G i and K i j obtained by replacing L i j by Σ i j also belong obviously to the commutant of {J 12 , J 34 , J 56 } but lead to trivial operators.

The Dirac model and Howe duality
We To that end, we shall introduce a Dirac model where P in (6) and osp(1|2) act as a dual reductive pair [19] on the eigenfunctions so that their respective irreducible representations can be paired through connections between the Casimir operators. The Dirac operator D as well as x and E are defined in six dimensions as follows: These operators have osp(1|2) as their dynamical algebra. Indeed, with the Z 2grading involution S given by and the presentation (2.1) of the osp(1|2) algebra is realized, with the total Casimir given by It should be noted that to any subset A ⊂ [6] there corresponds a realization of osp(1|2).
More precisely, if one defines these generators obey the osp(1|2) relations (2.1) with the involution given by when |A| is even. The Casimir has then the expression We will first couple the six representations pairwise (each pair will correspond to an osp(1|2) and we can hence use the previous observations for |A| even). We will refer to those pairs using the shortened index notation, j = 1, 2, 3. The j's will now label the representations we want to pair next, in this two-step process.
We observed that the Dirac equation in 4D provides an osp(1|2) made out ot two subsystems each realizing also an osp(1|2). It must hence result from the coproduct mapping: This connects with the construction of higher dimensional gamma matrices. Indeed, starting with a realization of the Clifford algebra Cℓ 2 (generated for example by the 2 γ i 's in (4.9)), a systematic way to construct a realization of a Clifford algebra in two additional dimensions (involving 4γ i 's) is [20] to takê This construction can be iterated as many times as needed for higher dimensional Clifford algebra realizations in even dimensions. With this choice of gamma matrices, the Dirac operator in 4D reads: which checks with the expected coproduct result. The algebra involution S is realized by the σ 3 matrix and its occurence is made manifest in this fashion.

Dimensional reduction
It is instructive to perform the dimensional reduction of the six-dimensional Dirac operator to R 3 . Introduce the cylindrical coordinates x 2 j−1 = ρ j cos θ j , We then have transformed expressions for D, x and E. In particular, We can now bring theγ µ 's back to their original form (the γ µ 's) by means of a rotation in spin space. Let a straightforward calculation shows that This rotation however leads to additional terms in the expression of D, which we can also eliminate with a gauge transformation depending on the radii and of the form e B = n i=1 f i (ρ i ). Requiring that after this additional transformation imposes that The following transformation with e B and S given by (5.7) and (5.4) respectively, is thus to be carried. The angular momentum J 12 (3.3) is one of the elements in the set whose commutant we looked for. It is transformed into and similar results hold for J 34 and J 56 . We also have Fixing J 2 j−1,2 j ∼ k j as a result of separation of variables, we can rewrite Note that these reduced operators still generate the same dynamical algebra since this is not altered by conjugation or separation of variables.
It is now interesting to investigate what is the effect of the reduction on the Casimirs operators. Recall that the reduced Casimir associated to the subset {i 1 , . . ., i n } = A ⊂ [6], with j ≡ {2 j − 1, 2 j}, is given by The reduced Casimirs: will satisfy the Bannai-Ito relations This should be compared with the system studied in [1], where the Dirac operator was given in terms of Dunkl derivatives D i of the Z 2 type The Bannai-Ito algebra was also seen to be the symmetry algebra in that case, with the one-index Casimirs Γ j equal to the deformation parameters k j . Here, we started with ordinary partial derivatives in twice as many dimensions and found the oneindex Casimirs taking the values of the angular momenta that are diagonalized in the dimensional reduction. Hence, the reduced system obtained here offers a new model with Bannai-Ito symmetry in addition to the Dirac-Dunkl one.

The higher rank Bannai-Ito algebra as a commutant
Let us now show how one can extend the result of the previous sections to the higher rank Bannai-Ito algebra B(n). Take any triple of pairwise disjoint subsets of [2n] called K , L, and M. There is an obvious isomorphism The Casimirs associated to one or two labels j are easily calculated with the help of formula (4.7) which remains valid for A ⊂ [2n]. One finds exactly the same operators Γ j and Γ i j with expressions as in (4.8) for i, j = 1, . . ., n.
These Casimir operators are immediately recognized as the generators of the commutant of the set {J 12 , . . ., J 2n−1,2n } in the spinorial representation of U (o(2n)).
Since we know that the intermediate Casimirs realize the commutation relations of the Bannai-Ito algebra, it is necessarily also the case for the generators of the commutant. This therefore confirms our claim to the effect that B(n) is the commutant of o(2) ⊕n .
We can again perform the dimensional reduction of the 2n-dimensional model to obtain a n-dimensional system whose symmetry algebra is B(n). Write D in the cylindrical coordinates (5.1) with j = 1, . . ., n and theγ µ 's defined as in (5.3). Exactly as was done in six dimensions, rotate theγ µ 's to their original form with the help of an operator S extending (5.4)  . Fixing J 2 j−1,2 j ∼ k j once the ignorable variable is eliminated, we can rewrite D as in (5.14) again extending the sum to n.
Note that the reduced operators still generate the same dynamical algebra. The reduction of the Casimirs is as described from (5.15) to (5.18), except that A is now a subset of [2n].
The reduced model thus obtained offers a new n-dimensional system in addition to the Dirac-Dunkl one, with the B(n) Bannai-Ito algebra as its symmetry algebra. The one-index Casimirs in the different models respectively take the values of the angular momenta and the deformation parameters in the Dunkl-derivatives.

An embedding of the R(n) Racah algebra in the B(n) Bannai-Ito algebra
The higher rank Racah algebra R(n) is an associative algebra with generators {C i , P i j } 1≤i< j≤n and defining relations [13]: [P kl , F i jk ] = P ik P jl − P il P jk , (7.1c) [F i jk , F jkl ] = F jkl P i j − F ikl P jk + 2C j − F i jk P jl , (7.1d) [F i jk , F klm ] = F ilm P jk − P ik F jlm , (7.1e) where i, j, k, l, m ∈ [n] are all different.
In [16] we identified the generators of the commutant of o(2) ⊕n (in oscillator representations of U (o(2n))) which are the invariants {G i , K i j } 1≤i< j≤n given in (3.12). With the following redefinitions a long but straightforward calculation in the oscillator realization showed that the defining relations (7.1) of the algebra R(n) were obeyed. It has been seen in [21] that R(3) admits an embedding in B(3). We already noted in Section 3 that the commutant picture brought this inclusion to the fore and we shall exploit it here together with the results in [16] to explicitly provide the embedding of R(n) in B(n).
The key point is that the intermediate Casimirs C i and C i j generating the R(n) algebra can be obtained from the intermediate sCasimirs of the B(n) algebra.

Conclusion
This paper has offered a novel presentation of the Bannai-Ito algebra B(n) as the commutant of o(2) ⊕ · · · ⊕ o(2) in the spinorial representation of o(2n) associated to the Clifford algebra Cℓ 2n . It has also indicated how this picture can be elegantly related to the definition of B(n) as the centralizer in U (osp(1|2) ⊗n ) of the coproduct embedding of the Lie superalgebra osp(1|2) in osp(1|2) ⊗n in the framework of the Howe duality associated to P in(2n) , osp(1|2) . This called for the introduction of a model involving the (massless) Dirac equation in 2n dimensions and identifying the connection between the Casimir operators that describe the paired representations of the two mutually commuting algebras on the solution space. Invariance under the subalgebra of P in(2n) allowed for dimensional reduction of the Dirac equation under separation of variables. This resulted in a model in R n with Bannai-Ito symmetry without reflection operators that hence differ from the Dirac-Dunkl equation already known to possess the same symmetry.
The commutant picture for B(n) made manifest the fact that the Racah algebra R(n) can be embedded in B(n). This observation had been made for n = 3 [21] and could here be explicitly extended.
Looking ahead it would be interesting to understand how various contractions of B(n) (and R(n)) play out within the commutant presentation. The relation with superintegrable systems would certainly be worth exploring [22,23].
The Askey-Wilson algebra [24] is similarly related to the quantum algebra U q (sl (2)). The rank 1 algebra encodes the bispectrality of the Askey-Wilson polynomials. Efforts are now deployed to construct the extensions to arbitrary ranks [25,26]. It is natural to think that the Askey-Wilson algebra also admits a dual commutant presentation. We plan on examining this matter which could shed useful light on the higher rank construction. We hope to report on these questions we have raised in the near future.