The q-Higgs and Askey–Wilson algebras

Abstract

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 o q 1/2 (2) ⊕ o q 1/2 (2) subalgebra of o q 1/2 (4) in the q-oscillator representation of the quantized universal enveloping algebra U q (u(4)). This q-Higgs algebra is also found as a specialization of the Askey-Wilson algebra embedded in the tensor product U q (su(1, 1)) ⊗ U q (su(1, 1)). The connection between these two approaches is established on the basis of the Howe duality of the pair o q 1/2 (4), U q (su(1, 1)) . 1 luc.frappat@lapth.cnrs.fr 2 gaboriaud@CRM.UMontreal.CA 3 eric.ragoucy@lapth.cnrs.fr 4 vinet@CRM.UMontreal.CA It is endowed with a Hopf structure with coproduct ∆ :We shall denote by U q (su(1, 1)) the non-compact real form of U q (sl 2 ) that has the three generators J ± and J 0 obeying J 0 , J ± = ±J ± , J − J + − q 2 J + J − = q 2J 0 2J 0 q .(1.4)

Howe duality and algebras of the Askey-Wilson type: an overview

“…The expressions of the operators K A and K B acting on H that realize the relations (8) (together with the specific central elements) are rather involved and we shall refer the reader to [8] for the formulas. We shall only stress that these operators can be obtained in a dual way: They are affinely related to the generators of the commutant of o q 1/2 (2) ⊕3 in o q 1/2 (6) as well as to the intermediate U q (su(1, 1)) Casimir elements C (1234) = ∆ (3) (C) ⊗ 1 ⊗ 1 and C (3456) = 1 ⊗ 1 ⊗ ∆ (3) (C) of the q-metaplectic representation (see (9)). This can be extended to higher ranks by letting n be arbitrary.…”

“…For n = 2 we are looking at the Clebsch-Gordan problen for U q (su(1, 1)). The q-Hahn algebra that arises has two dual realizations [9]: one as the commutant of o q 1/2 (2) ⊕2 in U q (u(4)) and the other in terms of the following two U q (su(1, 1)) operators, (∆(J 0 )⊗1⊗1)−(1⊗1⊗∆(J 0 )) and ∆ (2) (C) (the full Casimir element) in the q-metaplectic representation.…”

Hidden symmetry of Hahn problem for $sl_q(2)

“…The quadratic Hahn algebra QH ( 3) was shown to serve as a hidden symmetry in both quantum and classical pictures. Attempt of its q-generalization can be found in [4], but they were all limited only to the SU q (1, 1) case. Although it should be noted that in [5] a new addition rule is proposed for nonlinear algebras including sl q (2) ⊕ sl q (2) and two types of q-oscillator algebra.…”

The Askey-Wilson algebra and its avatars

“…Elements of representation theory have been investigated in [2,6,[18][19][20] and another of its manifestations is as a coideal subalgebra of U q (sl 2 ) [21][22][23]. The Askey-Wilson algebras have also been cast in the framework of Howe duality using the pair (U q (sl 2 ), o q 1/2 (2n)) [24][25][26][27]; they are special cases of the recently introduced Painlevé algebras [28] and belong to the Calabi-Yau class [29]. There is a significant connection to the field of algebraic combinatorics, as Askey-Wilson algebras are central in the classification of P -and Q-polynomial association schemes and the study of Leonard pairs and triples [30][31][32][33][34][35].…”