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

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

Hopf algebraic structure of the parabosonic and parafermionic algebras and paraparticle generalization of the Jordan Schwinger map

The aim of this paper is to show that there is a Hopf structure of the parabosonic and parafermionic algebras and this Hopf structure can generate the well known Hopf algebraic structure of the Lie algebras, through a realization of Lie algebras using the parabosonic (and parafermionic) extension of the Jordan Schwinger map. The differences between the Hopf algebraic and the graded Hopf superalgebraic structure on the parabosonic algebra are discussed.Comment: 11 pages, LaTex2e fil

New Solutions of the Yang-Baxter Equation Based on Root of 1 Representations of the Para-Bose Superalgebra Uq_q[osp(1/2)]

New solutions of the quantum Yang-Baxter equation, depending in general on three arbitrary parameters, are written down. They are based on the root of unity representations of the quantum orthosymplectic superalgebra \\U, which were found recently. Representations of the braid group $B_N$ are defined within any $N^{th}$ tensorial power of root of 1 \\U modules.Comment: 11 pages, PlainTe

Macroscopic properties of A-statistics

A-statistics is defined in the context of the Lie algebra sl(n+1). Some thermal properties of A-statistics are investigated under the assumption that the particles interact only via statistical interaction imposed by the Pauli principle of A-statistics. Apart from the general case, three particular examples are studied in more detail: (a) the particles have one and the same energy and chemical potential; (b) equidistant energy spectrum; (c) two species of particles with one and the same energy and chemical potential within each class. The grand partition functions and the average number of particles are among the thermodynamical quantities written down explicitly.Comment: 27 pages, 4 figures; to be published in J. Phys.

Unitarizable Representations of the Deformed Para-Bose Superalgebra Uq[osp(1/2)] at Roots of 1

The unitarizable irreps of the deformed para-Bose superalgebra $pB_q$, which is isomorphic to $U_q[osp(1/2)]$, are classified at $q$ being root of 1. New finite-dimensional irreps of $U_q[osp(1/2)]$ are found. Explicit expressions for the matrix elements are written down.Comment: 19 pages, PlainTe

Algebraic structure of the Green's ansatz and its q-deformed analogue

The algebraic structure of the Green's ansatz is analyzed in such a way that its generalization to the case of q-deformed para-Bose and para-Fermi operators is becoming evident. To this end the underlying Lie (super)algebraic properties of the parastatistics are essentially used.Comment: plain TeX, Preprint INRNE-TH-94/4, 13

Centre and Representations of U_q(sl(2|1)) at Roots of Unity

Quantum groups at roots of unity have the property that their centre is enlarged. Polynomial equations relate the standard deformed Casimir operators and the new central elements. These relations are important from a physical point of view since they correspond to relations among quantum expectation values of observables that have to be satisfied on all physical states. In this paper, we establish these relations in the case of the quantum Lie superalgebra U_q(sl(2|1)). In the course of the argument, we find and use a set of representations such that any relation satisfied on all the representations of the set is true in U_q(sl(2|1)). This set is a subset of the set of all the finite dimensional irreducible representations of U_q(sl(2|1)), that we classify and describe explicitly.Comment: Minor corrections, References added. LaTeX2e, 18 pages, also available at http://lapphp0.in2p3.fr/preplapp/psth/ENSLAPP583.ps.gz . To appear in J. Phys. A: Math. Ge

Jacobson generators, Fock representations and statistics of sl(n+1)

The properties of A-statistics, related to the class of simple Lie algebras sl(n+1) (Palev, T.D.: Preprint JINR E17-10550 (1977); hep-th/9705032), are further investigated. The description of each sl(n+1) is carried out via generators and their relations, first introduced by Jacobson. The related Fock spaces W_p (p=1,2,...) are finite-dimensional irreducible sl(n+1)-modules. The Pauli principle of the underlying statistics is formulated. In addition the paper contains the following new results: (a) The A-statistics are interpreted as exclusion statistics; (b) Within each W_p operators B(p)_1^\pm, ..., B(p)_n^\pm, proportional to the Jacobson generators, are introduced. It is proved that in an appropriate topology the limit of B(p)_i^\pm for p going to infinity is equal to B_i^\pm, where B_i^\pm are Bose creation and annihilation operators; (c) It is shown that the local statistics of the degenerated hard-core Bose models and of the related Heisenberg spin models is p=1 A-statistics.Comment: LaTeX-file, 33 page

The quantum superalgebra Uq[osp(1/2n)]U_q[osp(1/2n)]: deformed para-Bose operators and root of unity representations

We recall the relation between the Lie superalgebra $osp(1/2n)$ and para-Bose operators. The quantum superalgebra $U_q[osp(1/2n)]$, defined as usual in terms of its Chevalley generators, is shown to be isomorphic to an associative algebra generated by so-called pre-oscillator operators satisfying a number of relations. From these relations, and the analogue with the non-deformed case, one can interpret these pre-oscillator operators as deformed para-Bose operators. Some consequences for $U_q[osp(1/2n)]$ (Cartan-Weyl basis, Poincar\'e-Birkhoff-Witt basis) and its Hopf subalgebra $U_q[gl(n)]$ are pointed out. Finally, using a realization in terms of ``$q$-commuting'' $q$-bosons, we construct an irreducible finite-dimensional unitary Fock representation of $U_q[osp(1/2n)]$ and its decomposition in terms of $U_q[gl(n)]$ representations when $q$ is a root of unity.Comment: 15 pages, LaTeX (latex twice), no figure