A mapping construction for the q-deformed so(3) ⊂ u(3) embedding
Anticipating subsequent applications in nuclear structure theory, a first construction of a Dyson mapping for a q-deformed u(3) algebra, relevant to this field, is presented. To achieve this, a q-deformed sp(4, R) algebra is initially considered, realized in terms of tensor operators with respect to the standard suq (2) and containing a q-deformed so(3) angular momentum algebra. The desired mapping is then realized in terms of two boson-type conjugated tensor operators of first rank. A key problem is to determine the commutation relations between them. Our construction is based on the requirement that subsets of the commutation relations of the original so(3) algebra is preserved. As a result the images of the so(3)-subalgebra of sp(4, R) close the same commutation relations as the initial subalgebra of the angular momentum. In addition a q-deformed u(3) algebra, containing the so(3)-subalgebra of the images, is obtained. Its generators are the q-deformed components of a quadrupole operator, together with the images of the so(3)-subalgebra. In the limiting case q → 1 the reduction su(3) ⊃ so(3), crucial to nuclear structure physics, is recovered.