# Notes on $G_2$: The Lie algebra and the Lie group

## TL;DR

This paper provides a comprehensive, algebraic survey of the exceptional Lie group G2, including its Lie algebra, classifications, and relationships with spheres, with proofs and characterizations mainly at the Lie algebra level.

## Contribution

It offers a self-contained, detailed survey of G2's structure, classifications, and connections to spheres, with proofs and approaches not readily found in existing literature.

## Key findings

- Detailed classification of G2 Lie algebra and group
- Connections between G2 and spheres S^6 and S^7
- Various algebraic and geometric characterizations of G2

## Abstract

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 viewed and related.