The Direct Sum Decomposition of Type $G_2$ Lie Algebra (Communications in Mathematical Research, 2020)
This article mainly discusses the direct sum decomposition of type $G_2$ Lie algebra, which, under such decomposition, is decomposed into a type $A_1$ simple Lie algebra and one of its modules. Four theorems are given to describe this module, which could be the direct sum of two or three irreducible modules, or the direct sum of weight modules and trivial modules, or the highest weight module.
Keywords: simple Lie algebra $G_2$; simple Lie algebra $A_1$; direct sum decomposition; the highest weight module