Indecomposable representations and oscillator realizations of the exceptional Lie algebra G2

Abstract

In this paper various representations of the exceptional Lie algebra G2 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 G2, 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 Verma module are investigated in detail to construct the Dyson-Maleev-like realization of G2. After obtaining explicit forms of all twelve extremal vectors of the Verma module with the dominant highest weight $\Lambda $ , representations with their respective highest weights related to $\Lambda$ are systematically discussed. Moreover, this method can be used to construct fermion realizations from the irreducible representations, without referring to the Lie algebra chains, and a three-fermion realization is constructed as an example.