Fermionic realisations of simple Lie algebras

TL;DR

This paper explores fermionic realizations of simple Lie algebras using Dirac matrices, constructing irreducible representations on fermionic Fock spaces, and analyzing invariant fermionic operators with implications for quantum systems.

Contribution

It provides a general construction of fermionic realizations for any simple Lie algebra, including detailed cases for su(2), su(3), and su(5), and examines invariant fermionic operators derived from Lie algebra cohomology.

Findings

Constructed irreducible representations for su(2), su(3), and su(5).

Identified invariant fermionic operators related to Lie algebra cohomology.

Connected fermionic operators to the algebra's dimensionality and chirality operator.

Abstract

We study the representation ${\cal D}$ of a simple compact Lie algebra $\g$ of rank l constructed with the aid of the hermitian Dirac matrices of a (${\rm dim} \g$)-dimensional euclidean space. The irreducible representations of $\g$ contained in ${\cal D}$ are found by providing a general construction on suitable fermionic Fock spaces. We give full details not only for the simplest odd and even cases, namely su(2) and su(3), but also for the next (${dim} \g$)-even case of su(5). Our results are far reaching: they apply to any $\g$-invariant quantum mechanical system containing ${\rm dim} \g$ fermions. Another reason for undertaking this study is to examine the role of the $\g$-invariant fermionic operators that naturally arise. These are given in terms of products of an odd number of gamma matrices, and include, besides a cubic operator, (l-1) fermionic scalars of higher order. The latter are constructed from the Lie algebra cohomology cocycles, and must be considered to be of theoretical significance similar to the cubic operator. In the (${\rm dim} \g$)-even case, the product of all l operators turns out to be the chirality operator $\gamma_q, q=({{\rm dim} \g+1})$.