The equitable presentation of the quantum superalgebra osp q (1|2), in which all generators appear on an equal footing, is exhibited. It is observed that in their equitable presentations, the quantum algebras osp q (1|2) and sl q (2) are related to one another by the formal transformation q → −q. A q-analog of the Bannai-Ito algebra is shown to arise as the covariance algebra of osp q (1|2).
AMS classification numbers: 17B37, 20G42, 81R50
IntroductionThe purpose of this Letter is threefold: to display the equitable, or Z 3 -symmetric, presentation of the quantum superalgebra osp q (1|2), to show that the equitable presentations of osp q (1|2) and sl q (2) are related to one another by the formal transformation q → −q, and to demonstrate that the covariance algebra of osp q (1|2) is a q-analog of the Bannai-Ito algebra.Our considerations take root in the Racah problem for the su(2) algebra, i.e. the coupling of three angular momenta. In this problem, the states are usually described in terms of the quantum numbers j i associated to the individual angular momenta J i with i ∈ {1, 2, 3}, the quantum number j associated to the total angular momentum J = J 1 + J 2 + J 3 , the quantum number M associated to the projection of the total angular momentum J along one axis, and any one of the quantum numbers j 12 , j 23 , j 31 associated to the intermediate angular momenta (23), (31)}. These bases are related via Racah coefficients [3]. The main drawback of such bases is their involved behavior under particle permutations. To circumvent this problem, Chakrabarti [2], Lévy-Leblond and Lévy-Nahas [20] devised an "equitable" coupling scheme and showed that there is a "democratic" basis specified by the quantum numbers j 1 , j 2 , j 3 , j and ζ, where ζ is the eigenvalue of the volume operator ∆ = ( J 1 × J 2 ) · J 3 . The three angular momenta J i enter symmetrically in this scheme and the states of the democratic basis have definite behaviors under particle permutations.The Racah-Wilson algebra is the hidden algebraic structure behind the Racah problems of su(2) and su(1, 1) [11]. The concern for a democratic approach to these Racah problems leads to the equitable presentation of this algebra [9]. In this presentation, the defining relations of the RacahWilson algebra are Z 3 -symmetric and all the generators appear on an equal footing, whence the epithets "equitable" or "democratic". It was recently shown in [5] that the equitable generators of the Racah/Wilson algebra can also be realized as quadratic expressions in the equitable sl(2) generators proposed in [14]. Note that the Racah-Wilson algebra also arises as symmetry algebra for superintegrable systems [16] and encodes the bispectrality of the Racah/Wilson polynomials [6].