Abstract :
A Lie algebra is said to be split graded if it is graded by a torsion free abelian group Q in such a way that the subalgebra 0 is abelian and the operators ad 0 are diagonalized by the grading. The elements of Q%{0} with α ≠ {0} are called roots and a root α is said to be integrable if there are root vectors x± α ± α which are ad-nilpotent and generate an 2-subalgebra (α). In this paper we study subalgebras Π generated by the subalgebras (α), α Π, where Π is a set of integrable roots. For Π ≤ 2, these subalgebras are essentially Kac–Moody algebras which permits us to generalize several results on root strings from Kac–Moody algebras to split graded algebras. A central result is the local finiteness theorem saying that whenever all roots of a split graded Lie algebra μ are integrable, then μ is locally finite. If differences of roots in Π are not roots, then Π is called a simple system. In this case we describe the structure of the subalgebras Π in their relationship to the corresponding Kac–Moody algebras.
