Deformed CliffordCl_q(n m) and orthosymplecticU_q[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 Cl q (n|m) such that the latter is a homomorphic image of the quantum superalgebra U q [osp(2n + 1|2m)]. The Fock space F (n|m) of Cl q (n|m) is constructed. At q being a root of unity (q = exp(iπl/k)) q−bosons (and q−fermions) are operators acting in a finite-dimensional subspace F l/k (n|m) of F (n|m). Each F l/k (n|m) is turned through the above mentioned homomorphism into an irreducible (root of unity) U q [osp(2n + 1|2m)] module. For q being a primitive root of unity (l = 1) the corresponding representation is unitary. The module F 1/k (n|m) is decomposed into a direct sum of irreducible U q [sl(m|n)] submodules. The matrix elements of all Cartan-Weyl elements of U q [sl(m|n)] are given within each such submodule.

Hopf structure and Green ansatz of deformed parastatistics algebras

“…We have adopted the notaion π(x) = x and use N i = h i ∓ 1 2 . The analysis [22] of the positivity of the norm for the pB q (n) and pF q (n) representations in the simplest case p = 1 shows that such unitary representations (realized as finite dimensional factor representions) exist only for q being a root of unity.…”

A quantum fermion realisation of the finite dimensional spinor representation of Uq(osp(1|2n))

“…In order to generalise the construction to U q (osp(2m+1|2n)) for m > 0, it is necessary to bring quantum bosons into the picture. The quantum fermions and bosons together form a deformed Clifford superalgebra Cl q (m|n), the structure of which has been studied extensively in the literature (see, e.g., [3]). However, the deformed Clifford superalgebra is used quite differently in this paper (see Section 2.1 and Definition 3.1).…”

“…The quantum boson-fermion realisation of U q (osp(2m + 1|2n), Θ 2 ) and quantum fermion-boson realisation of U q (osp(2m + 1|2n), Θ 1 ) were studied in [4], while [3] constucted the irreducible representations of U q (osp(2m + 1|2n), Θ 2 ). In this paper, our definitions of U q (osp(2m + 1|2n), Θ 1 ) and Cl q (m, n) are different from the set up of [3], though our approach is similar to theirs. This leads to a new realisation of U q (2m + 1|2n, Θ 1 ) (see Theorem 2.5 and Theorem 3.2) different from those obtained before.…”