Kac-Moody Lie algebras graded by Kac-Moody root systems

TL;DR

This paper studies how Kac-Moody Lie algebras can be decomposed into graded components using root systems, extending existing notions to more general cases and classifying gradations for certain types.

Contribution

It generalizes the concept of C-admissible pairs to all Kac-Moody Lie algebras and constructs specific subalgebras that induce finite gradings, providing a classification framework.

Findings

Constructed C-admissible subalgebras for general Kac-Moody algebras.

Established classification of gradations for affine and hyperbolic types.

Introduced generalized C-admissible pairs for indefinite types.

Abstract

We look to gradations of Kac-Moody Lie algebras by Kac-Moody root systems with finite dimensional weight spaces. We extend, to general Kac-Moody Lie algebras, the notion of C-admissible pair as introduced by H. Rubenthaler and J. Nervi for semi-simple and affine Lie algebras. If g is a Kac-Moody Lie algebra (with Dynkin diagram indexed by I) and (I,J) is such a C-admissible pair, we construct a C-admissible subalgebra g^J, which is a Kac-Moody Lie algebra of the same type as g, and whose root system \Sigma grades finitely the Lie algebra g. For an admissible quotient \rho : I \rightarrow I we build also a Kac-Moody subalgebra g^\rho which grades finitely the Lie algebra g. If g is affine or hyperbolic, we prove that the classification of the gradations of g is equivalent to those of the C-admissible pairs and of the admissible quotients. For general Kac-Moody Lie algebras of indefinite type, the situation may be more complicated; it is (less precisely) described by the concept of generalized C-admissible pairs.