The $q$-Higgs and Askey-Wilson algebras

A $q$-analogue of the Higgs algebra, which describes the symmetry properties of the harmonic oscillator on the $2$-sphere, is obtained as the commutant of the $\mathfrak{o}_{q^{1/2}}(2) \oplus \mathfrak{o}_{q^{1/2}}(2)$ subalgebra of $\mathfrak{o}_{q^{1/2}}(4)$ in the $q$-oscillator representation of the quantized universal enveloping algebra $U_q(\mathfrak{u}(4))$. This $q$-Higgs algebra is also found as a specialization of the Askey--Wilson algebra embedded in the tensor product $U_q(\mathfrak{su}(1,1))\otimes U_q(\mathfrak{su}(1,1))$. The connection between these two approaches is established on the basis of the Howe duality of the pair $\big(\mathfrak{o}_{q^{1/2}}(4),U_q(\mathfrak{su}(1,1))\big)$.


Introduction
The Higgs algebra was first obtained by Higgs [1] as the algebra of the conserved quantities of the Coulomb problem and harmonic oscillator on the 2-sphere. Shown to be isomorphic to the Hahn algebra [2], it was also identified as the symmetry algebra of the Hartmann potential [3], of certain ring-shaped potentials [4] and of the singular oscillator in two dimensions [5,6]. The Higgs algebra stands between Lie algebras and quantized universal enveloping algebras, as it can be viewed both as a deformation of the su(2) Lie algebra [7] and a truncation of the U q (sl 2 ) quantum algebra [8]. It has been obtained as the quantum finite W-algebra W (sp(4), 2 sl(2)) [9,10] and has also appeared in the context of Heisenberg quantization of identical particles [11].
The Higgs algebra can be presented in the following form 1 The U q (su(1, 1)), o q (n) algebras and their q-oscillators realizations The duality connection that we shall invoke in our discussion involves the algebras U q (su(1, 1)) and o q (n). We shall thus begin by introducing these algebras and their q-oscillator realizations.
Let q be a complex number such that |q| < 1. One defines for any number x the following q-numbers: (1.1) The same notation will be used for operators.
We introduce next the non-standard q-deformation o q (n) of o(n) which is defined as the associative unital algebra with generators L i,i+1 (i = 1, . . . , n − 1) and relations In the litterature, this non-standard deformation is often denoted U ′ q (so n ), see for instance [34][35][36][37]. It has been shown in [38] that o q (n) can be viewed as a q-analogue of the symmetric space based on the pair (gl(n), o(n)). Although it has no Hopf structure on its own, it is a coideal subalgebra of U q (sl(n)) [38] and appears in many areas of mathematical physics [36].
The two cases where n = 3 and n = 4 are especially of interest to us.
Let us first note that it is possible to consider a so-called "Cartesian" presentation [39][40][41] of U q (sl 2 ), in which the three generators play an "equitable" role, and which corresponds to the non-standard deformation o q (3) (equivalently U ′ q (so 3 ) in refs. [40,41]) of the universal enveloping algebra U (so(3)), obtained by modifying the defining relations for the skew-symmetric generators of so(3).
It goes like this. With j 0 , j ± , the U q (sl 2 ) generators, form the following elements: where {a, b} = ab + ba is the anticommutator and g is a normalization factor. Defining ba is the q-commutator, j 1 , j 2 and j 3 then satisfy the "Cartesian" relations Upon identifying L 12 = j 1 , L 23 = j 2 , one finds that this corresponds precisely to the relations (1.7) for the algebra o q (3). Note that the relations (1.7c) do not exist in this case.
For what follows, it will also be useful to have the formulas for o q (4) in full. These relations read [42] L 12 L 2 23 − (q + q −1 )L 23 L 12 L 23 + L 2 23 L 12 = −L 12 , (1.10a) It is immediate to see that L 12 , L 23 and L 23 , L 34 respectively generate two o q (3) subalgebras of o q (4), however they do not appear within a direct sum, in contrast to what happens with o(4).
If one introduces the following elements: where [a, b] q is defined as above and [a, b] q −1 := q − 1 2 ab − q 1 2 ba, the two independent Casimir operators of the algebra o q (4) are then given by [27,34,41] (1.12b)

The q-oscillator algebras, Schwinger and metaplectic realizations
Let us now recall the properties of the q-oscillator operators that will be used to realize the algebras presented above. The q-oscillator algebra A q (n) [43][44][45] is defined as the unital associative algebra over C generated by n independent sets of q-oscillators 13) and such that the commutators between elements with different indices i are equal to zero. The last two relations allow one to express (1.14) In the limit q → 1, A 0 i coincides with the usual number operator N i . The q-oscillator algebra has the following representation on the space spanned by the standard occupancy number states |n 1 , · · · , n n = |n 1 ⊗ · · · ⊗ |n n (n i ∈ N): These commuting q-oscillators can now be used to build realizations of the algebras considered above.
Firstly, the algebra o q (3) can be realized à la Schwinger in terms of two q-oscillators. More precisely, using the homomorphism χ : U q (sl 2 ) → A q (2) given by and the identification (1.8), the following realization of o q (3) is obtained: Another key ingredient is the metaplectic realization of U q (su(1, 1)), which is given by the homomorphism µ : U q (su(1, 1)) → A q (1): One sees immediately that it is a q-deformation of the usual metaplectic representation of su(1, 1).
Finally, we shall also use the realization of o q 1/2 (4) in terms of 4 q-oscillators which is provided by: (1.20) One checks that the L i,i+1 indeed verify relations of the form (1.10) but whose q's have been replaced by q 1/2 's. Furthermore, L 12 , L 34 commute and hence generate a (4)) and the q-Higgs algebra It was shown in [13] that the Higgs algebra appears as the commutant of o(2) ⊕ o(2) in the universal enveloping algebra U (u (4)). This section aims to define the q-Higgs algebra through a q-analogue of this commutant picture.
Introduce the following three operators which commute with the generators L 12 and L 34 (in the limit q → 1, L 12 and L 34 correspond to rotations in the (1, 2) and (3, 4) planes).
One notes that each big parenthesis in the expression of the M ± operators can actually be obtained by applying the coproduct of U q (su(1, 1)) to the J ± generators. Recalling that the bilinears of the form E ij = A + i A − j , i, j = 1, 2, 3, 4 realize the U q (u(4)) algebra [46], it can be observed that M ± , L generate the non-trivial part of the commutant of o q 1/2 (2) ⊕ o q 1/2 (2) in the q-oscillator realization of U q (u(4)).
It is immediate to see that M ± and L also commute with the central element One could ask how were the expressions for L, M ± obtained. First, the operator L obviously commutes with L 12 and L 34 . Second, instead of obtaining the factors in M ± from the coproduct one can look for elements T ± in A q (2) that commute with L 12 ; this is most easily done "on-shell", that is, by solving [L 12 , T ± ]|n 1 , n 2 = 0 for any two q-oscillator states. One thus arrives at Since A 0 1 + A 0 2 = 1 2 (L + H), only the second factor of T ± is relevant. The same is done with L 34 on the direct product states |n 3 , n 4 . It is then clear that the only combinations of the operators (2.3) and their (3,4) analogues that will belong to U q (u(4)) are those occurring in M ± .
It now remains to determine the algebra formed by the three generators M ± and L.

Proposition 2.1
The operators M ± and L have the following commutators: (2.4a) The elements L 12 , L 34 and H, which are central, play the role of structure constants. We shall take these relations to define abstractly the (universal) q-Higgs algebra. (2.4b) Proof: The first relations of (2.4a) are obvious. The last relation is obtained by a direct computation in the q-oscillator algebra. Starting with (2.1a)-(2.1b), and using the identity Now, from the expression (1.20), one obtains and a similar expression for L 34 2 with the replacement and after some algebra, one is left with the following equation (2.9) Expressing the A 0 i generators in terms of L and H, one finally obtains the desired commutation relation.
Remark 2.2 In the limit q → 1, noting that one easily recovers from (2.4a) the commutation relations of the Higgs algebra (0.1) in the form: Hence, the relations (2.4) indeed define a q-deformation of the Higgs algebra.
3 The Askey-Wilson algebra and an embedding into U q (su(1, 1)) ⊗2 We now indicate how (a special case of) the Askey-Wilson algebra can be embedded in the tensor product U q (su(1, 1)) ⊗ U q (su(1, 1)). With ∆ the coproduct of U q (su(1, 1)) given in (1.5), we can take where C denotes the Casimir operator given in (1.6).
Defining K 3 = K 1 , K 2 , a direct calculation gives We now proceed to calculate the commutation relations of K 1 , K 2 , K 3 . They are seen to take the form of the relations of the Askey-Wilson (AW) algebra which read where r is as in (3.3) below and ξ 1 , . . . , ξ 7 are arbitrary in the generic AW situation.
After a rather cumbersome calculation, using the expressions (3.1) for the K i 's as well as the commutation relations (1.4), one finds that the K i 's indeed obey the relations (3.2) with the following specific expressions for the parameters: where C (1) = C ⊗ 1 and C (2) = 1 ⊗ C are respectively the Casimir operators in the spaces 1 and 2 of the tensor product, and J (12) 0 = ∆(J 0 ). These quantities C (i) and J (12) 0 commute with K 1 , K 2 and K 3 and we hence have a version of (3.2) that is centrally extended.
Since there are only three independant quantities entering the ξ i 's (there are four in the general case), we conclude that the K 1 , K 2 , K 3 generate a specialization of the Askey-Wilson algebra. One checks that in the limit q → 1, the parameters r, ξ 2 , ξ 3 vanish and one recovers the Hahn algebra. The standard q-Hahn algebra is obtained from the Askey-Wilson algebra by setting for instance ξ 1 = 0 in (3.2). The algebra satisfied by K 1 , K 2 and K 3 is actually isomorphic to the q-Hahn algebra as the standard form of the latter [28,47] is obtained by taking K 2 = K 2 − ξ 1 /r. The limit q → 1 is singular however if we adopt this presentation.
4 The q-Higgs algebra, the Askey-Wilson algebra, and the dual pair o q 1/2 (4) , U q (su(1, 1)) We shall explain in this section how the q-Higgs algebra obtained as a commutant and the special Askey-Wilson algebra found from the embedding just described are connected through Howe duality and are in fact isomorphic.
Take 4 metaplectic representations defined as in (1.19). We will add them first pairwise using the U q (su(1, 1)) coproduct (1.5): Mindful of Section 2, it is immediate to check that where the L i,i+1 are defined as in (1.20). Let us stress that (4.3) makes the key statement that the algebras U q (su(1, 1)) and o q 1/2 (4) are mutually commuting in the q-oscillator realization.
It has been shown [27] that o q 1/2 (4) and U q (su(1, 1)) actually form a Howe dual pair. (They constitute precisely the quantum analogue of the classical pair (o(4), su(1, 1)) which was used in the analysis of the Higgs and Hahn algebras [13].) This means that their representations can be connected through their Casimirs. We will now proceed to indicate explicitly how this is realized.
To that end, we first put the J (2i−1,2i) • in correspondance with the J • from Section 3. Let us focus on the coproduct embeddings (4.1). As each pairing of U q (su(1, 1)) in the spaces (1,2) and (3,4) gives a copy of U q (su(1, 1)), we can embed the specialization of the Askey-Wilson algebra of Section 3 into these two copies of U q (su(1, 1)).
Also note that the quantities C (1) and C (2) are the images of the Casimir operators C (1) and C (2) and that they are directly related to the L 12 and L 34 by In view of this, it is now evident that in the q-oscillator realization, the generators of the special Askey-Wilson algebra are expressible in terms of those of the q-Higgs algebra, and vice-versa. Hence these two algebras are isomorphic, as in the q → 1 case.
To wrap things up, let us point out that the two Casimirs of o q 1/2 (4) given in (1.12) have a direct interpretation in this q-oscillator framework.
The first Casimir of o q 1/2 (4), denoted C 4 , corresponds to the total Casimir of the quadruple tensor product of U q (su(1, 1)): This is precisely the pairing of the Casimirs of o q 1/2 (4) and U q (su(1, 1)) that follows from the Howe duality.
The second Casimir of o q 1/2 (4), denoted C ′ 4 , is identically zero in the q-oscillator realization: It can be seen as the q-analogue of the usual relation between the angular momenta, see for instance Let us mention in closing this section that the q → 1 limit of the above yields straightforwardly the duality presented in [13] between the Higgs or the Hahn algebras viewed as a commutant in U (u(4)) or embedded in U (su(1, 1)) ⊗ U (su(1, 1)).

Conclusion
Summing up, we have introduced a q-analogue of the Higgs algebra by looking for the commutant of a o q 1/2 (2) ⊕ o q 1/2 (2) subalgebra of o q 1/2 (4) in the q-oscillator representation of U q (u(4)). This algebra was then seen to be isomorphic to a special case of the Askey-Wilson algebra (itself isomorphic to the standard q-deformation of the Hahn algebra) which has an embedding in U q (su(1, 1)) ⊗ U q (su (1, 1)). The Howe dual pair o q 1/2 (4), U q (su(1, 1)) was then invoked as the reason behind this double picture.
The q-oscillator realization in which o q 1/2 (4) and U q (su(1, 1)) commute can be generalized easily for o q 1/2 (n) with n arbitrary. It is known that o q 1/2 (n), U q (su (1, 1)) is a dual pair [27]. This opens up the door to the study of the full Askey-Wilson algebra. We hypothesize that it should be possible to obtain this algebra as the commutant of a o q 1/2 (2) ⊕ o q 1/2 (2) ⊕ o q 1/2 (2) subalgebra of o q 1/2 (6) in U q (u(6)) in this q-oscillator representation. It would be also of high interest to see if the higher rank Askey-Wilson algebras [49,50] could be obtained in a similar fashion.
It should moreover be mentioned that the dual pair o q 1/2 (n), U q (su(1, 1)) was analyzed in [27] in a q-commuting variable framework. It would be quite interesting to see if some sort of dimensional reduction in q-commuting variables could be performed to obtain a q-analogue of the superintegrable model on the n-sphere [51]. We hope to address all these questions in the near future.