The Higgs and Hahn algebras from a Howe duality perspective
Introduction
This paper is concerned with the Higgs algebra [1], [2], [3] and the discrete version of the Hahn algebra [4], [5] which actually designate two different but isomorphic presentations of the same algebra [3], [6]. We aim to establish that this algebra arises as the commutant of a subalgebra of in the oscillator representation of the universal algebra . We will moreover point out that this relation that the Higgs and Hahn algebras have with is in duality, in the sense of Howe [7], [8], [9], with the one they are known to have with [10], [11]. Let us start with some background.
The Hahn algebra has three generators , and subjected to the relations where and a, b, , , , are structure constants. We assume (otherwise (1.1) would be equivalent to the Lie algebra ). This algebra describes the eigenvalue problems of both the discrete and continuous Hahn polynomials [12]. We shall henceforth consider the discrete case where and which is realized by the bispectral operators of the Hahn polynomials (see [13] for instance). In this case, upon performing the affine transformation one can cast the commutation relations in the form with , constants (or central elements).
The Hahn algebra admits an embedding in that we shall describe in details later as it is germane to our analysis. This observation underscores its connection to the Clebsch-Gordan problem for (and ).
The Higgs algebra can be viewed as a polynomial deformation of . It has three generators D, , satisfying the following commutation relations: with , central elements. That the Higgs algebra is isomorphic to the discrete Hahn algebra is readily seen by taking and observing that the commutation relations (1.3) then follow from (1.4) with Historically, the algebra defined in (1.4) was found by Higgs, hence the name, as the one realized by the conserved quantities of the Coulomb problem and harmonic oscillator on the two-sphere. It can be viewed as a deformed algebra [14] or a truncation of the quantum algebra [6]. This algebra has been identified as the symmetry algebra of the Hartmann [4] and of certain ring-shaped potentials [5] as well as the singular oscillator in two dimensions [2], [3]. The Higgs algebra has moreover emerged in the Heisenberg quantization of identical particles [15]. Furthermore, it has been seen to coincide with the finite quantum W-algebra [16], [17]. (For a review of finite W-algebras and their applications, see [18].)
Similarly to the Hahn case, the Racah algebra [3], [10], [19] is realized by the bispectral operators of the corresponding polynomials. It admits an embedding in with the intermediate Casimir elements representing the generators. The Hahn algebra can be obtained through a contraction of the standard presentation of the Racah algebra in a way that parallels the limit that takes the Racah polynomials into those of Hahn [12]. A generalization of the Racah algebra to higher ranks is found in [20].
Recently the Racah algebra has been interpreted in a Howe duality framework and shown to be a commutant [21] in the enveloping algebra of , the Lie algebra of the rotation group in six dimensions. An extension of this result to the generalized Racah algebra is given in [22]. An analogous treatment of the Bannai-Ito algebra [23], [24], [25], which is in a sense a supersymmetric version of the Racah algebra, was also achieved in [26]. These advances raised the question of how to describe the Higgs algebra from a Howe duality perspective. The answer to this question will be provided here with the significant merit of expanding and interconnecting the various descriptions of the Higgs and Hahn algebras.
The remainder of the paper is organized as follows. As preparation background, familiar results on the metaplectic representation of and the embedding of in the Heisenberg-Weyl algebra will be reviewed in Section 2. The Higgs algebra will be obtained as the commutant of in in Section 3. The embedding of the Hahn algebra into will be described in Section 4. The two pictures of the Higgs/Hahn algebra presented in Sections 3 and 4 will be connected via the Howe dual pair that acts on the state vectors of the four-dimensional oscillator. Dimensional reduction will be used in Section 6 to recover the fact that the symmetries of the singular oscillator in two dimensions generate the Higgs algebra. The paper will end with a summary of the findings and an outlook.
Section snippets
, and oscillators
We shall be dealing with the Heisenberg-Weyl algebra generated by n pairs of oscillator operators , , , that satisfy The number operators are such that In the position coordinates , these operators read The Lie algebra has generators , , obeying the commutation relations Its Casimir element is given by
The Higgs algebra as a commutant in
We are now ready to obtain our first main result, namely that the Higgs algebra can be defined as a commutant. Pick the subalgebra of generated by and ; clearly . We want to concentrate on the commutant of this subalgebra in . We are thus looking for polynomials in the generators , that are invariant under rotations in both the - and -planes. It is not difficult to convince oneself that an integrity basis for that set is provided
The embedding of the Hahn algebra into
Let us here indicate how the Hahn algebra is embedded in the tensor product of with itself. Let be the coproduct homomorphism with in the superscript notation introduced in Section 2. Consider the following identification [10], [11]: that is, is the image of the Casimir element under the coproduct. It is clear that the computation of the overlaps coefficients between the
The Howe duality connection
We shall now indicate that the two descriptions of the Hahn algebra presented in Section 3 and 4 can be connected through Howe duality. It is known (see in particular [9]) that there is a pairing between the representations of and that act in a mutually commuting way (see (5.3)) on the state space of the four-dimensional harmonic oscillator. We shall exploit this to show that the embedding of the Hahn algebra in the double tensor product of the universal enveloping algebra of one
Dimensional reduction and the singular oscillator in two dimensions
We shall now carry the dimensional reduction of the four-dimensional isotropic harmonic oscillator under the action to identify in this way the Higgs/Hahn symmetry of the singular oscillator in the plane.
Make the change of variables Eliminate the 's by separating the variables with Take the eigenvalues of this operator equal to . After performing the gauge transformation one sees that the
Conclusion
This paper has provided a synthetic description of the Higgs and Hahn algebras in light of Howe duality. With the understanding that the Higgs and the (discrete) Hahn algebras are isomorphic, we have shown that this algebra can be viewed as a commutant in . It has also been recalled that it can be embedded in the tensor product of with itself. The two approaches have been linked in view of the fact that and form a dual pair on the state space of the harmonic
Acknowledgements
The authors would like to thank E. Ragoucy and P. Sorba for informative discussions. LV wishes to acknowledge the hospitality of the CNRS and of the LAPTh in Annecy where part of this work was done. JG holds an Alexander-Graham-Bell scholarship from the Natural Science and Engineering Research Council (NSERC) of Canada. The research of LV is supported in part by a Discovery Grant from NSERC. SV enjoys a Neubauer No Barriers scholarship at the University of Chicago and benefitted from a Metcalf
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