Open Access
2012 Relations Among Low-Dimensional Simple Lie Groups
Robert Gilmore
J. Geom. Symmetry Phys. 28: 1-45 (2012). DOI: 10.7546/jgsp-28-2012-1-45


The compact classical Lie groups can be regarded as groups of $n \times n$ matrices over the real, complex, and quaternion fields $\mathbb{R,~ C},$ and $\mathbb{Q}$ that satisfy metric- and volume-conserving conditions. These groups, ${\rm SO}(n,\mathbb{R}), {\rm SU}(n,\mathbb{C})$, and ${\rm Sp}(n,\mathbb{Q})$, are not all independent. Homomorphisms exist among some of these groups for small dimension. We review these relations by describing the Lie algebras of the compact forms and their complex extensions. Other noncompact real forms of these Lie algebras are constructed by systematic methods. The relations among all distinct real forms is presented.


Download Citation

Robert Gilmore. "Relations Among Low-Dimensional Simple Lie Groups." J. Geom. Symmetry Phys. 28 1 - 45, 2012.


Published: 2012
First available in Project Euclid: 26 May 2017

zbMATH: 1317.22005
MathSciNet: MR3114813
Digital Object Identifier: 10.7546/jgsp-28-2012-1-45

Rights: Copyright © 2012 Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences

Back to Top