ℤ-Gradations of Classical Affine Lie Algebras and Kac Parameters

Abstract

The main result of this paper is an algorithm which determines the Kac parameters for every ℤ-gradation of the type ˜s of classical affine Lie algebras, i.e., Lie algebras with Dynkin diagrams of the type

We would like to emphasize that the algorithm determines also the corresponding basis of the root system of the classical affine Lie algebra. Such a basis we name the Kac basis.

Key Words:

• ℤ-Gradations of a Kac–Moody algebras
• Classical affine Lie algebras
• Kac parameters of finite order automorphisms of simple Lie algebras

Mathematics Subject Classification:

• 17B67(17B70)

Acknowledgments

The author expresses his gratitude to his adviser Prof. Mirko Primc for pointing out this problem and for many useful discussions on the subject. The author also expresses his gratitude to an anonymous referee for helpful suggestions and especially for a shorter proof of the main Theorem 3.6.

Notes

^#Communicated by K. Misra