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arXiv · 1202.3347

Indecomposable representations and oscillator realizations of the exceptional Lie algebra G_2

Hua-Jun Huang·You-Ning Li·Dong Ruan
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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.

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BibTeXRIS

Hua-Jun Huang, You-Ning Li, Dong Ruan. 2012-02-15. Indecomposable representations and oscillator realizations of the exceptional Lie algebra G_2. https://arxiv.org/abs/1202.3347

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