If is a split Lie algebra, which means that is a Lie algebra with a root decomposition = + ฮฑโ ฮฑ , then the roots of can be classified into different types: a root ฮฑ โ is said to be of nilpotent type if all subalgebras
x ฮฑ x -ฮฑ = span x ฮฑ x -ฮฑ x ฮฑ x -ฮฑ for x ยฑฮฑ โ ยฑฮฑ are nilpotent, and of simple type if there exist elements x ยฑฮฑ โ ยฑฮฑ such that x ฮฑ x -ฮฑ โผ = 2
. A simple root ฮฑ โ is called integrable if there exist elements x ยฑฮฑ โ ยฑฮฑ such that x ฮฑ x -ฮฑ โผ = 2 and the endomorphisms ad x ยฑฮฑ are locally nilpotent (Section I).
The role of integrable roots in split Lie algebras has been investigated by K.-H. Neeb in [Ne98]. One important result of this paper is the Local Finiteness Theorem which states that a split Lie algebra with only integrable roots is locally finite, i.e., the Lie algebra is the direct limit of its finite dimensional subalgebras.
In this paper we focus from the outset on locally finite split Lie algebras. Our objective is to describe the correspondence between the root types of and the structural features of a locally finite split Lie algebra .
If is finite dimensional, then has a unique -invariant Levi decomposition where the radical as well as the -invariant Levi complement can be described in terms of root types (Theorem II.1). One of the main results of this paper is an analog of this statement for locally finite split Lie algebras, saying that a locally finite split Lie algebra has a generalized Levi decomposition. This means that โผ = i i where is the unique maximal locally solvable ideal of , is an -invariant semisimple subalgebra of that is generated by the root spaces of integrable roots, and is a subspace of the abelian Lie algebra (Theorem III.16). The existence of a 664