Indecomposable representations and oscillator realizations of the exceptional Lie algebra G_2

TL;DR

This paper explores algebraic representations of the exceptional Lie algebra G_2, providing explicit multi-boson and multi-fermion realizations, and systematically analyzing elementary and related representations.

Contribution

It introduces new oscillator realizations of G_2, including six-boson, five-boson, and three-fermion models, and explicitly determines matrix elements of the master representation.

Findings

Explicit six-boson realization of elementary G_2 representations

Construction of a new three-fermion realization from the fundamental representation

Systematic analysis of extremal vectors and related representations

Abstract

In this paper various representations of the exceptional Lie algebra G_2 are investigated in a purely algebraic manner, and multi-boson/multi-fermion realizations are obtained. Matrix elements of the master representation, which is defined on the space of the universal enveloping algebra of G_2, are explicitly determined. From this master representation, different indecomposable representations defined on invariant subspaces or quotient spaces with respect to these invariant subspaces are discussed. Especially, the elementary representations of G_2 are investigated in detail, and the corresponding six-boson realization is given. After obtaining explicit forms of all twelve extremal vectors of the elementary representation with the highest weight {\Lambda}, all representations with their respective highest weights related to {\Lambda} are systematically discussed. For one of these representations the corresponding five-boson realization is constructed. Moreover, a new three-fermion realization from the fundamental representation (0,1) of G_2 is constructed also.