Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 027, 4 pages      arXiv:1412.4721      https://doi.org/10.3842/SIGMA.2015.027

An Integrability Condition for Simple Lie Groups II

Maung Min-Oo
Department of Mathematics & Statistics, McMaster University, Hamilton, Canada

Received December 17, 2014, in final form March 26, 2015; Published online April 01, 2015

Abstract
It is shown that a simple Lie group $G$ ($\neq {\rm SL}_2$) can be locally characterised by an integrability condition on an $\operatorname{Aut}(\mathfrak{g})$ structure on the tangent bundle, where $\operatorname{Aut}(\mathfrak{g})$ is the automorphism group of the Lie algebra of $G$. The integrability condition is the vanishing of a torsion tensor of type $(1,2)$. This is a slight improvement of an earlier result proved in [Min-Oo M., Ruh E.A., in Differential Geometry and Complex Analysis, Springer, Berlin, 1985, 205-211].

Key words: simple Lie groups and algebras; $G$-structure.

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