Relations among low-dimensional simple Lie groups. (English) Zbl 1317.22005

Gilmore, Robert

J. Geom. Symmetry Phys. 28, 1-45 (2012).

The groups of \(n\times n\) matrices \(\mathrm{SO}(n,{\mathbb R})\), \(\mathrm{SU}(n,{\mathbb C})\) and \(\mathrm{SO}(n,{\mathbb Q})\) are the standard examples of the classical Lie groups over the real \({\mathbb R}\), complex \({\mathbb C}\) and quaternion fields \( {\mathbb Q}\). At low dimensions among them there exist some important isomorphisms or homomorphisms. Based on their Lie algebra structure the paper under review describes these relations by identifying the respective groups by their dimension and rank. Especially, the role of the real forms is emphasized and clarified. In fact, this article should be considered as a nice compendium to all books dealing with the structures of the Lie algebras, Lie groups and their representations.

Reviewer: Clementina Mladenova (Sofia)

MSC:

22E15	General properties and structure of real Lie groups
22E10	General properties and structure of complex Lie groups
17B20	Simple, semisimple, reductive (super)algebras

Keywords:

Lie groups; Lie algebras; isomorphisms; homomorphisms; representations; parameterizations; rotations; Lorentz transformations; Heisenberg nilpotent group; Casimir invariants and covariants