Jacobson generators of the quantum superalgebra Uq[sl(n+1∣m)]U_q[sl(n+1|m)] and Fock representations

As an alternative to Chevalley generators, we introduce Jacobson generators for the quantum superalgebra $U_q[sl(n+1|m)]$. The expressions of all Cartan-Weyl elements of $U_q[sl(n+1|m)]$ in terms of these Jacobson generators become very simple. We determine and prove certain triple relations between the Jacobson generators, necessary for a complete set of supercommutation relations between the Cartan-Weyl elements. Fock representations are defined, and a substantial part of this paper is devoted to the computation of the action of Jacobson generators on basis vectors of these Fock spaces. It is also determined when these Fock representations are unitary. Finally, Dyson and Holstein-Primakoff realizations are given, not only for the Jacobson generators, but for all Cartan-Weyl elements of $U_q[sl(n+1|m)]$.Comment: 27 pages, LaTeX; to be published in J. Math. Phy

Fock representations of the superalgebra sl(n+1|m), its quantum analogue U_q[sl(n+1|m)] and related quantum statistics

Fock space representations of the Lie superalgebra $sl(n+1|m)$ and of its quantum analogue $U_q[sl(n+1|m)]$ are written down. The results are based on a description of these superalgebras via creation and annihilation operators. The properties of the underlying statistics are shortly discussed.Comment: 12 pages, PlainTex; to appear in J. Phys. A: Math. Ge

Microscopic and macroscopic properties of A-superstatistics

The microscopic and the macroscopic properties of A-superstatistics, related to the class A(0,n-1)\equiv sl(1|n) of simple Lie superalgebras are investigated. The algebra sl(1|n) is described in terms of generators f_1^\pm, >..., f_n^\pm, which satisfy certain triple relations and are called Jacobson generators. The Fock spaces of A-superstatistics are investigated and the Pauli principle of the corresponding statistics is formulated. Some thermal properties of A-superstatistics are constructed under the assumption that the particles interact only via statistical interaction imposed by the Pauli principle. The grand partition function and the average number of particles are written down explicitly in the general case and in two particular examples: 1) the particles have one and the same energy and chemical potential; 2) the energy spectrum of the orbitals is equidistant.Comment: 26 pages, 3 figure

A Superalgebra Morphism of Uq[OSP(1/2N)] onto the Deformed Oscillator Superalgebra Wq(N)

We prove that the deformed oscillator superalgebra $W_q(n)$ (which in the Fock representation is generated essentially by $n$ pairs of $q$-bosons) is a factor algebra of the quantized universal enveloping algebra $U_q[osp(1/2n)]$. We write down a $q$-analog of the Cartan-Weyl basis for the deformed $osp(1/2n)$ and give also an oscillator realization of all Cartan-Weyl generators.Comment: 8 pages, PlainTeX, University of Ghent, Dept. Math. Int. Report TWI-93-1