N. I. Stoilova

The

We define the quadratic algebra su(2)_{\alpha} which is a one-parameter deformation of the Lie algebra su(2) extended by a parity operator. The odd-dimensional representations of su(2) (with representation label j, a positive integer) can be extended to representations of su(2)_{\alpha}. We investigate a model of the finite one-dimensional harmonic oscillator based upon this algebra su(2)_{\alpha}. It turns out that in this model the spectrum of the position and momentum operator can be computed explicitly, and that the corresponding (discrete) wavefunctions can be determined in terms of Hahn polynomials. The operation mapping position wavefunctions into momentum wavefunctions is studied, and this so-called discrete Hahn-Fourier transform is computed explicitly. The matrix of this discrete Hahn-Fourier transform has many interesting properties, similar to those of the traditional discrete Fourier transform.

\mathfrak {su}(2)_\alpha

We define the quadratic algebra su(2)_{\alpha} which is a one-parameter deformation of the Lie algebra su(2) extended by a parity operator. The odd-dimensional representations of su(2) (with representation label j, a positive integer) can be extended to representations of su(2)_{\alpha}. We investigate a model of the finite one-dimensional harmonic oscillator based upon this algebra su(2)_{\alpha}. It turns out that in this model the spectrum of the position and momentum operator can be computed explicitly, and that the corresponding (discrete) wavefunctions can be determined in terms of Hahn polynomials. The operation mapping position wavefunctions into momentum wavefunctions is studied, and this so-called discrete Hahn-Fourier transform is computed explicitly. The matrix of this discrete Hahn-Fourier transform has many interesting properties, similar to those of the traditional discrete Fourier transform.

Hahn oscillator and a discrete Fourier–Hahn transform

An orthogonal basis of weight vectors for a class of infinite-dimensional representations of the orthosymplectic Lie superalgebra osp(2m+1|2n) is introduced. These representations are particular lowest weight representations V(p), with a lowest weight of the form [-p/2,...,-p/2|p/2,...,p/2], p being a positive integer. Explicit expressions for the transformation of the basis under the action of algebra generators are found. Since the relations of algebra generators correspond to the defining relations of m pairs of parafermion operators and n pairs of paraboson operators with relative parafermion relations, the parastatistics Fock space of order p is also explicitly constructed. Furthermore, the representations V(p) are shown to have interesting characters in terms of supersymmetric Schur functions, and a simple character formula is also obtained.

The su(2)_{\alpha} Hahn oscillator and a discrete Hahn-Fourier transform

An orthogonal basis of weight vectors for a class of infinite-dimensional representations of the orthosymplectic Lie superalgebra osp(2m+1|2n) is introduced. These representations are particular lowest weight representations V(p), with a lowest weight of the form [-p/2,...,-p/2|p/2,...,p/2], p being a positive integer. Explicit expressions for the transformation of the basis under the action of algebra generators are found. Since the relations of algebra generators correspond to the defining relations of m pairs of parafermion operators and n pairs of paraboson operators with relative parafermion relations, the parastatistics Fock space of order p is also explicitly constructed. Furthermore, the representations V(p) are shown to have interesting characters in terms of supersymmetric Schur functions, and a simple character formula is also obtained.

A class of infinite-dimensional representations of the Lie superalgebra osp(2m + 1|2n) and the parastatistics Fock space

The goal of this paper is to give an explicit construction of the Fock spaces of the parafermion and the paraboson algebra, for an infinite set of generators. This is equivalent to constructing certain unitary irreducible lowest weight representations of the (infinite rank) Lie algebra so(\infty) and of the Lie superalgebra osp(1|\infty). A complete solution to the problem is presented, in which the Fock spaces have basis vectors labelled by certain infinite but stable Gelfand-Zetlin patterns, and the transformation of the basis is given explicitly. We also present expressions for the character of the Fock space representations.

A class of infinite-dimensional representations of the Lie superalgebra and the parastatistics Fock space

We introduce and obtain multimode paraboson coherent states. In appropriate subspaces these coherent states provide a decomposition of unity where the measure, when expressed using the cat-type states, is positive definite. Bicoherent states where the mutually commuting lowering operators are diagonalized are also obtained. Matrix elements in the coherent state basis are calculated.

PARAFERMIONS, PARABOSONS AND REPRESENTATIONS OF 𝔰𝔬(∞) AND 𝔬𝔰𝔭(1|∞)

As an alternative to Chevalley generators, we introduce Jacobson generators for the quantum superalgebra Uq[sl(n+1|m)]. The expressions of all Cartan–Weyl elements of Uq[sl(n+1|m)] in terms of these Jacobson generators become very simple. We determine and prove certain triple relations between the Jacobson generators, necessary for a complete set of supercommutation relations between the Cartan–Weyl elements. Fock representations are defined, and a substantial part of this paper is devoted to the computation of the action of Jacobson generators on basis vectors of these Fock spaces. It is also determined when these Fock representations are unitary. Finally, Dyson and Holstein–Primakoff realizations are given, not only for the Jacobson generators, but for all Cartan–Weyl elements of Uq[sl(n+1|m)].

Representations of the orthosymplectic Lie superalgebra \mathfrak{osp}(1|4) and paraboson coherent states

We introduce a new Gelfand-Zetlin (GZ) basis for covariant representations of gl(n|n). The patterns in this basis are fixed according to a chain of subalgebras, all of which are Lie superalgebras themselves. The basic generators consist of odd elements only. This GZ basis is interesting because the limit when n goes to infinity becomes clear. This could be used in the description of systems with an infinite number of parabosons and parafermions.

Jacobson generators of the quantum superalgebra Uq[sl(n+1|m)] and Fock representations

The spinor representations of the orthosymplectic Lie superalgebras osp(m|n) are considered and constructed. These are infinite-dimensional irreducible representations, of which the superdimension coincides with the dimension of the spinor representation of so(m-n). Next, we consider the self dual tensor representations of osp(m|n) and their generalizations: these are also infinite-dimensional and correspond to the highest irreducible component of the

The “odd” Gelfand-Zetlin basis for representations of general linear Lie superalgebras

p^{th}