
Appendix D
Definition and Basic Facts of Lie Algebras
In this appendix we recall the definition and some basic facts of Lie algebras which can
be found in any textbook on Lie algebras, for instance, [585]. We also recommend the two
books [452, 461] having a more physics-oriented presentation.
Recall that an algebra a is a vector space endowed with a bilinear map · : a × a → a;
in the following, all considerations will be over C. An algebra is called a Lie algebra if the
bilinear operation – then called the Lie bracket and denoted by “[·, ·]” – satisfies two special
properties.
Definition D.1 A Lie algebra g is an algebra such that the map [·, ·]:g ×g →