N ov 2 00 0 Deformations of the Boson sp ( 4 , R ) Representation and its Subalgebras

The boson representation of the sp(4, R) algebra and two distinct deformations of it, spq (4, R) and spt(4, R), are considered, as well as the compact and noncompact subalgebras of each. The initial as well as the deformed representations act in the same Fock space, H, which is reducible into two irreducible representations acting in the subspaces H + and H − of H. The deformed representation of spq (4, R) is based on the standard q-deformation of the boson creation and annihilation operators. The subalgebras of sp(4, R) (compact u(2) and noncompact u ε (1, 1) with ε = 0, ±) are also deformed and their deformed representations are contained in spq (4, R). They are reducible in the H + and H − spaces and decompose into irreducible representations. In this way a full description of the irreducible unitary representations of uq (2) of the deformed ladder series u 0 q (1, 1) and of two deformed discrete series u ± q (1, 1) are obtained. The other deformed representation, spt(4, R), is realized by means of a transformation of the q-deformed bosons into q-tensors (spinor-like) with respect to the suq (2) operators. All of its generators are deformed and have expressions in terms of tensor products of spinor-like operators. In this case, a deformed sut(2) appears in a natural way as a subalgebra and can be interpreted as a deformation of the angular momentum algebra so(3). Its representation in H is reducible and decomposes into irreducible ones that yields a complete description of the same. The basis states in H + , which require two quantum labels, are expressed in terms of three of the generators of the sp(4, R) algebra and are labeled by three linked integer parameters.