Bulg. J. Phys. vol.35 no.s1 (2008), pp. 403-414

Unitary Representations of the Lie Superalgebra osp(1|2n) and Parabosons

S. Lievens1, N.I. Stoilova1,2, J. Van der Jeugt1
1Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281-S9, B-9000 Gent, Belgium
2Institute for Nuclear Research and Nuclear Energy, Boul. Tsarigradsko Chaussee 72, 1784 Sofia, Bulgaria
Abstract. It is known that there is a close connection between the Fock space of n pairs of boson operators B±i (i = 1, 2, ..., n) and the so-called metaplectic representation V(1) of the Lie superalgebra osp(1|2n) with lowest weight (1/2,1/2,...,1/2). On the other hand, the defining relations of osp(1|2n) are equivalent to the defining relations of n pairs of paraboson operators b±i. In particular, with the usual star conditions, this implies that the "parabosons of order p" correspond to a unitary irreducible (infinite-dimensional) lowest weight representation V(p) of osp(1|2n) with lowest weight (p/2,p/2,...,p/2). Apart from the simple cases p = 1 or n = 1, these representations had never been constructed due to computational difficulties, despite their importance.
We have now managed to give an explicit and elegant construction of these representations V(p), and can present explicit actions or matrix elements of the osp(1|2n) generators. Essentially, V(p) is constructed as a quotient module of an induced module. In all steps of the construction and for the chosen basis vectors, the subalgebra u(n) of osp(1|2n) plays a crucial role.

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