Lie algebraic aspects of quantum statistics. parafermi statistics

We study in detail the Lie-algebraical properties of the quantization condition for spinor fields

$\text{[a}{}^{\mathrm{+}}{}_{\mathrm{i}}\text{,a}{}^{\mathrm{?}}{}_{\mathrm{j}}\text{],a}{}^{\mathrm{\pm }}{}_{\mathrm{i}}\text{= ±2δ}{}_{\mathrm{ij}}\text{a}{}^{\mathrm{\pm }}{}_{\mathrm{j}}$

.It turns out that the parafermi statistics is one particular solution of this relation: it is the minimal rank simple Lie algebra generated by the operators a^±_i entering into the above relation. We point out some other solutions.     

Highest weight representations of the Lie algebra gl∞

Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 24, No. 1, pp. 82–83, January–March, 1990.     

Parastatistics and the lie algebraical methods used there

We review some of the properties of the parafield operators and discuss in some detail where the difference between the ordinary and paraquantisation originates. Particular attention is paid to the many-vacuum representations of the para-Fermi operators.     

A3-Weight multiplicity formula

We write down an explicite formula for the multiplicity of a weight in an arbitrary irreducible module of the classical algebra A₃. For this purpose we first derive a reccurrence relation between the multiplicities of the weights in A_a and A_s?1.     

Causal A-statistics I. General properties

It is shown that the second quantization axioms can, in principle, be satisfied by creation and annihilation operators generating the Lie algebra of the unimodular group. The Fock spaces W(p,q) are labelled with two arbitrary non-negative numbers p and q. The Pauli principles are formulated. In the Fock space W(p,q) there can not be more than p+q particles in a single state. The charge of an arbitrary ensemble of particles can not exceed p and be less than q.     

An isomorphism between the Para-Fermi algebra and the algebra O(n,n+1)

It is shown that the relations which define the para-Fermi creation and annihilation operatorsb _i ,a _i (i = 1,...,n) can be considered as commutation relations of the algebra 0(n, n+1). It turns out that every representation of 0(n, n+1) determines a representation of the para-Fermi algebra and vice versa.