Coxeter-Killing Transformations of Simple Lie Algebras

Abstract

To determine the Betti numbers of complex simple Lie groups (algebras) is a classical problem. It is known that these Betti numbers are simply the coefficients of the Poincaré polynominal

$$p\left( t \right)=\left( 1+{{t}^{{{2}_{1}}m+1}} \right)\cdots \left( 1+{{t}^{2{{m}_{1}}+1}} \right)$$

of the corresponding Lie group, where m ₁,..., m ₁ are called the Poincaré indices of the Lie algebras. Brauer and Pontryagin (see Ref. 1) already determined these numbers for four classes of classical simple Lie algebras, and in 1950 C. Chevalley [1] announced the result for several exceptional Lie algebras.