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Notes on $G_2$: The Lie algebra and the Lie group. (arXiv:1704.07819v1 [math.RA])
来源于:arXiv
These notes have been prepared for the Workshop on "(Non)-existence of
complex structures on $\mathbb{S}^6$", to be celebrated in Marburg in March,
2017. The material is not intended to be original. It contains a survey about
the smallest of the exceptional Lie groups: $G_2$, its definition and different
characterizations joint with its relationship with $\mathbb{S}^6$ and with
$\mathbb{S}^7$. With the exception of the summary of the Killing-Cartan
classification, this survey is self-contained, and all the proofs are given,
mainly following linear algebra arguments. Although these proofs are
well-known, they are spread and some of them are difficult to find. The
approach is algebraical, working at the Lie algebra level most of times. We
analyze the complex Lie algebra (and group) of type $G_2$ as well as the two
real Lie algebras of type $G_2$, the split and the compact one. Octonions will
appear, but it is not the starting point. Also, 3-forms approach and spinorial
approach are viewe 查看全文>>