A q-Deformation of the Parastatistics and an Alternative to the Chevalley Description of $U_{q}[osp(2n+1/2m)]$

Abstract

The paper contains essentially two new results. Physically, a deformation of the parastatistics in a sense of quantum groups is carried out. Mathematically, an alternative to the Chevalley description of the quantum orthosymplectic superalgebra U q [osp(2n + 1/2m)] in terms of m pairs of deformed parabosons and n pairs of deformed parafermions is outlined.

Parastatistics Algebra, Young Tableaux and the Super Plactic Monoid

“…The parastatistics algebras (4) of creation and annihilation operators allow for q-deformations as introduced by Palev [21]. The idea is to replace the universal enveloping algebra (UEA) U (osp 1+2m|2n ) by the quantum UEA U q (osp 1+2m|2n ) written in an alternative form, with a system of relations between generators corresponding to the parastatistics creation and annihilation operators.…”

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

“…Although the quantization (q-deformation) of simple Lie algebras and basic Lie superalgebras is usually carried out in terms of their Chevalley generators, there exist alternative descriptions in terms of so-called deformed creation and annihilation operators for the q-deformation of osp(1|2n) [24], so(2n + 1) [25], osp(2n + 1|m) [19], sl(n + 1) [26] and sl(n + 1|m) [27]. These alternative generators have the advantage that in some natural interpretation they have a direct physical significance; furthermore, they allow the definition and construction of a mathematically interesting and physically important class of irreducible representations, the Fock representations.…”

“…This is another reason to call them Jacobson generators (JG's). The link between the JG's and the simple Lie superalgebras provides a natural background for their q-deformations (we refer to [19] for more discussion in this respect).…”

Jacobson generators, Fock representations and statistics of sl(n+1)

“…One such possibility is to consider deformations of parastatistics, namely deformations of so(2n + 1) and osp(1/2n) in the sense of quantum groups. We refer to [65] for discussions and results along this line.…”