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

TL;DR

This paper explores the properties of A-statistics related to sl(n+1) Lie algebras, describing their generators, Fock spaces, and connections to exclusion statistics, with new results on operator limits and model statistics.

Contribution

It introduces a novel interpretation of A-statistics as exclusion statistics and analyzes the limits of Jacobson generator operators, linking them to Bose operators.

Findings

A-statistics are interpreted as exclusion statistics.

Operators B(p)_i^\pm converge to Bose creation/annihilation operators as p→∞.

Local statistics of certain models correspond to p=1 A-statistics.

Abstract

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.