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

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

S. Lievens

go back^{1}, N.I. Stoilova^{1,2}, J. Van der Jeugt^{1}^{1}*Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281-S9, B-9000 Gent, Belgium*^{2}*Institute 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 ofnpairs of boson operatorsB(^{±}_{i}i = 1, 2, ..., n) and the so-calledmetaplecticrepresentationV(1)of the Lie superalgebraosp(1|2n)with lowest weight(1/2,1/2,...,1/2). On the other hand, the defining relations ofosp(1|2n)are equivalent to the defining relations ofnpairs of paraboson operatorsb. In particular, with the usual star conditions, this implies that the "parabosons of order^{±}_{i}p" correspond to a unitary irreducible (infinite-dimensional) lowest weight representationV(p)ofosp(1|2n)with lowest weight(p/2,p/2,...,p/2). Apart from the simple casesp = 1orn = 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 representationsV(p), and can present explicit actions or matrix elements of theosp(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 subalgebrau(n)ofosp(1|2n)plays a crucial role.