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Euclidean Lie Algebras

Published online by Cambridge University Press:  20 November 2018

Robert V. Moody
Robert V. Moody*
Affiliation:
University of Saskatchewan, Saskatoon, Saskatchewan
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Our aim in this paper is to study a certain class of Lie algebras which arose naturally in (4). In (4), we showed that beginning with an indecomposable symmetrizable generalized Cartan matrix (A ij) and a field Φ of characteristic zero, we could construct a Lie algebra E((A ij)) over Φ patterned on the finite-dimensional split simple Lie algebras. We were able to show that E((A ij)) is simple providing that (A ij) does not fall in the list given in (4, Table). We did not prove the converse, however.

The diagrams of the table of (4) appear in Table 2. Call the matrices that they represent Euclidean matrices and their corresponding algebras Euclidean Lie algebras. Our first objective is to show that Euclidean Lie algebras are not simple.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 1432 - 1454
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Coxeter, H. S. M., Regular polytopes, 2nd ed. (Macmillan, New York, 1963).Google Scholar
2. Iwahori, N., On the structure of the Hecke ring of a Chevalley group over a finite field, J. Fac. Sci. Univ. Tokyo (1) 10 (1964), 215–236.Google Scholar
3. Jacobson, N., Lie algebras (Interscience, New York, 1962).Google Scholar
4. Moody, R. V., A new class of Lie algebras, J. Algebra 10 (1968), 211–230.Google Scholar