Root Fernando–Kac subalgebras of finite type

Abstract

Let g be a finite-dimensional Lie algebra and M be a g -module. The Fernando–Kac subalgebra of g associated to M is the subset g [ M ] ⊂ g of all elements g ∈ g which act locally finitely on M. A subalgebra l ⊂ g for which there exists an irreducible module M with g [ M ] = l is called a Fernando–Kac subalgebra of g . A Fernando–Kac subalgebra of g is of finite type if in addition M can be chosen to have finite Jordan–Hölder l -multiplicities. Under the assumption that g is simple, I. Penkov has conjectured an explicit combinatorial criterion describing all Fernando–Kac subalgebras of finite type which contain a Cartan subalgebra. In the present paper we prove this conjecture for g ≄ E 8 .