Coxeter-Killing Transformations of Simple Lie Algebras

  • Hung-Hsi Wu
Part of the The University Series in Mathematics book series (USMA)


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.


Positive Root Weyl Group Betti Number High Root Dynkin Diagram 
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  1. 1.
    C. Chevalley, The Betti numbers of the exceptional simple Lie groups, in Proc. Int. Congress of Mathematicians, II, pp. 21–24 (1951).Google Scholar
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    A. J. Coleman, The Betti numbers of the simpler Lie groups, Canad. J. Math. 10, (1958).Google Scholar
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    H. S. M. Coxeter, The product of the generators of a finite group generated by reflections, Duke. Math. J. 18, (1951).Google Scholar
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    B. Kostant, Am. J. Math. 81, 973–1032 (1959).MathSciNetMATHCrossRefGoogle Scholar
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    JI. H. Manses, O nonynpocTbix nojrpynnax rpynnJIH. AH CCCP. 8, (1944).Google Scholar
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    Wan Ze-xian, Lie algebras, Science Press, Beijing (1964) (in Chinese).Google Scholar
  7. 7.
    H. Weyl, The Classical Groups, Princeton University Press, Princeton (1946).MATHGoogle Scholar

Copyright information

© Plenum Press, New York 1991

Authors and Affiliations

  • Hung-Hsi Wu
    • 1
  1. 1.University of CaliforniaBerkeleyUSA

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