The quantum superalgebra U_q(osp(1/2n)): deformed para-Bose operators and root of unity representations

Abstract

We recall the relation between the Lie superalgebra osp(1/2n) and para-Bose operators. The quantum superalgebra U q [osp(1/2n)], defined as usual in terms of its Chevalley generators, is shown to be isomorphic to an associative algebra generated by so-called pre-oscillator operators satisfying a number of relations. From these relations, and the analogue with the non-deformed case, one can interpret these pre-oscillator operators as deformed para-Bose operators. Some consequences for U q [osp(1/2n)] (Cartan-Weyl basis, Poincaré-Birkhoff-Witt basis) and its Hopf subalgebra U q [gl(n)] are pointed out. Finally, using a realization in terms of "q-commuting" q-bosons, we construct an irreducible finitedimensional unitary Fock representation of U q [osp(1/2n)] and its decomposition in terms of U q [gl(n)] representations when q is a root of unity. Short title: Quantum superalgebra U q [osp(1/2n)]

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

“…25. A similar problem for U q [osp(1/2m)], corresponding to a q−deformation of the pB operators, was first carried out for m = 1 26 and then for any m. 27,28,29 Here we generalize the results for any U q [osp(2n + 1/2m)], n, m > 1, namely when both para-Bose and para-Fermi operators are involved. This amounts to a simultaneous deformation of the parabosons and the parafermions as one single supermultiplet.…”

Unitarizable representations of the deformed para-Bose superalgebra Uq[osp(1/2)] at roots of 1

“…Using the approach of the present paper one can try to construct representations (including root of 1 representations) for pB q (n) = U q [osp(1/2n)]. To this end one can use n-pairs of deformed pB operators as given in [2,4,5]. The solution, however, is not going to be easy for arbitrary values of p, if one takes into account that the problem has not been solved even in the nondeformed case.…”

“…. , a ± n are not more involved [4,5]. At n = 1, namely in the case we consider, a ± are proportional to the Chevalley generators of pB q = U q [osp(1/2)].…”

Variants of bosonization in parabosonic algebra: the Hopf and super-Hopf structures in parabosonic algebra

“…Starting in the early '80 's, and using the recent (by that time) results in the classification of the finite dimensional simple complex Lie superalgebras which was obtained by Kac (see: [12,13] but also [19]), Palev managed to identify the parabosonic algebra with the universal enveloping algebra of a certain simple complex Lie superalgebra. In [9], [32] and [31], Palev shows the following:…”