Abstract
Superalgebras and supersymmetries are relevant to mixed systems of bosons and fermions. A simple systematic procedure to construct the multi-level unitary and orthosymplectic superalgebras which appear in the generalized multi-level pairing models and for arbitrary choice of level degeneracies is presented. The algebraic substructures of these superalgebras are discussed and the explicit formulae for their generators and Casimir operators are given in spherical tensor (Racah) form. We draw attention to the existence of four general types of dynamical supersymmetry, including two vibrational limits and two γ-unstable limits. Their analytical expressions of the energies are derived. The relations between pairing operators and the multipole operators for the supersymmetric multi-level system are given in a closed form. The results presented show very clearly the duality between the orthosymplectic superalgebras and the generalized quasi-spin algebras for the supersymmetric system with pairing interactions. Using the concept of complementary subalgebras, the full super two-level pairing Hamiltonian, with uniform pairing strength, and the dynamical symmetries are rewritten, in a more convenient form. The generalization of the class C transitional Hamiltonian is demonstrated.