Deformed CliffordClq(n m) and orthosymplecticUq[osp(2n 1 2m)] superalgebras and their root of unity representations

Abstract

It is shown that the Clifford superalgebra Cl(n|m) generated by m pairs of Bose operators (odd elements) anticommuting with n pairs of Fermi operators (even elements) can be deformed to Clq(n|m) such that the latter is a homomorphic image of the quantum superalgebra Uq[osp(2n + 1|2m)]. The Fock space F(n|m) of Clq(n|m) is constructed. For q being a root of unity (q = exp(i?l/k)) q-bosons (and q-fermions) are operators acting in a finite-dimensional subspace Fl/k(n|m) of F(n|m). Each Fl/k(n|m) is turned through the above-mentioned homomorphism into an irreducible (root of unity) Uq[osp(2n + 1|2m)] module. For q being a primitive root of unity (l = 1) the corresponding representation is unitary. The module F1/k(n|m) is decomposed into a direct sum of irreducible Uq[sl(m|n)] submodules. The matrix elements of all Cartan?Weyl elements of Uq[sl(m|n)] are given within each such submodule.