Roots of Kac-Moody Lie algebras

Abstract

The notion of Kac-Moody Lie algebras has recently been introduced and studied as a natural generalization of a finite dimensional split semisimple Lie algebra with successful applications to Macdonald type identities(cf. Lepowsky [3]). Such a Lie algebra has a root system, which is a natural analogue of the usual root systems in the sense of Bourbaki [1]. In this paper, we will give a characterization of positive root systems of Kac-Moody Lie algebras as a subset of a lattice. (See Proposition 1 and Theorem 1 below.) Let A be a generalized Cartan matrix and L the Kac-Moody Lie algebra associated with A, and let A (resp. A+) be the root system (resp. the positive root system) of L (for the definition,see §1). In §2, we will consider the special positive root system P(A) associated with L. This system P(A) satisfiesthe conditions (XI), (X2), (Yl), (Y2) and (Y3), which are specifiedin §2. Conversely we will show that any set satisfying these conditions coincides with the system P(A) arising from some Kac-Moody Lie algebra. In particular, A+ is uniquely determined by (Yl), (Y2) and (Y3) when A is given. On the other hand, there are two kinds of roots in A, called real roots and imaginary roots respectively. In §3, we will present a characterization of imaginary roots. In §4, we will give a way to produce the roots of L inductively