Finite Cartan matrices and Lie algebras of classical type

Abstract

The Mills-Seligman result is that a classical Lie algebra L is a direct sum of simple ideals, each of which is again classical; that the simple classical Lie algebras are determined by their Cartan matrices; and that the possible indecomposable Cartan matrices are given by the usual classification, i.e., are of types A,,, B, ,..., G,. The proof of this result, in [3], is surprisingly hard. The difficulty arises from the fact that the root system of L is contained in H*, a vector space over F, while on the other hand, the Cartan matrix must be a matrix over H. Deducing information about the Cartan matrix (over Z) from information about the root system (over F) is difftcult. (See, for example, Lemmas 115.3 and 11.5.4 of [3].) In this paper we give a simpler proof of the Mills-Seligman theorem. Our technique is to associate to L a certain subset, A, of H’ which plays the role of root system and apply standard characteristic zero results (of [ 11) to A. These transference methods were used before by Seligman (see [4]), but only after much more theory in characteristic p was developed.