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 finite-dimensional 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.

15 pages, LaTeX (latex twice), no figures

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• Fock representations of the superalgebra sl(n+1|m), its quantum analogue U_q[sl(n+1|m)] and related quantum statistics
• A description of the quantum superalgebra $U_q[sl(n+1|m)]$ via creation and annihilation generators
• Unitarizable Representations of the Deformed Para-Bose Superalgebra Uq[osp(1/2)] at Roots of 1
• Deformed Clifford Cl_q(n|m) and orthosymplectic U_q[osp(2n+1|2m)] superalgebras and their root of unity representations
• Bosonisation and Parastatistics: An Example and an Alternative Approach
• Highest weight irreducible representations of the Lie superalgebra $gl(1/\infty )$
• Jacobson generators of the quantum superalgebra $U_q[sl(n+1|m)]$ and Fock representations
• New Solutions of the Yang-Baxter Equation Based on Root of 1 Representations of the Para-Bose Superalgebra U${}_q$[osp(1/2)]
• Parabosonic and parafermionic algebras. Graded structure and Hopf structures
• Variants of bosonisation in Parabosonic algebra. The Hopf and super-Hopf structures