Integrable Roots in Split Graded Lie Algebras

✍ Scribed by Karl-Hermann Neeb

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 314 KB
• Volume: 225
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1999.8108

✦ Synopsis

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 adnilpotent and generate an 2 -subalgebra α . In this paper we study subalgebras ⊆ generated by the subalgebras α , α ∈ , where is a set of integrable roots. For ≤ 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 are integrable, then is locally finite. If differences of roots in are not roots, then is called a simple system. In this case we describe the structure of the subalgebras in their relationship to the corresponding Kac-Moody algebras.