STRUCTURES OF WEYL GROUPS OF REAL SEMISIMPLE LIE ALGEBRAS

Abstract

Let g be a real semisimple Lie algebra, h a Cartan subalgebra of g, g~c and h~c the respective complexifications of g and h, and a the conjugation of g with respect to g. Denote by W(h~c) the Weyl group of g~c acting on h~c, and write W_σ(h) to be the subgoup of W(hc) with elements leaving g invariant, and then W(h) indicates the group of restrictions to h of inner automorphisms of g leaving h invariant. W(h) and W_σ(h) can be called the Weyl group and the quasi-Weyl group of g with respect to h respectively. In this paper, we give a clear expression to the strueture of groups W(h) and W_σ(h) for every Cartan subalgebra h of g (see Theorem 5). The above two groups are calculated in detail for every class of the Cartan subalgebras of the classical simple Lie algebra g (see Table 1).