Maximal Subalgebras of Kac–Moody Algebras

Abstract

We use the classification of finite order automorphisms by Kac to characterize all maximal subalgebras, regular, semisimple, reductive or not of a simple complex Lie algebra (up to conjugacy) that we can determine from its Dynkin diagram. Using Barnea et al. [Barnea, Y., Shalev, A., Zelmanov, E. I. (1998). Graded subalgebras of affine Kac–Moody algebras. Israel J. Math. 104:321–334] we extend our results to the case of affine Kac–Moody algebras. We also point out some inaccuracies in the Dynkin paper [Dynkin, E. B. (1957a). Semisimple subalgebras of semisimple Lie algebras. Amer. Math. Soc. Transl t. 6:111–244].

Key Words:

• Maximal subalgebras
• Dynkin diagrams
• Kac–Moody algebras

AMS Subject Classification (2000) :

• 17B40
• 17B20
• 17B67

Acknowledgment

The first author is very grateful to Professors J. Alev and Mme M. P. Malliavin for hospitality in their team at Paris VI University.

Notes

^#Communicated by J. Alev.