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Algorithmic construction of Chevalley bases

Part of: Linear algebraic groups and related topics Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  01 December 2012

K. Magaard  and
R. A. Wilson
K. Magaard
Affiliation:
School of Mathematics, University of Birmingham, Birmingham B15 2TT, United Kingdom (email: [email protected])
R. A. Wilson
Affiliation:
School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, United Kingdom (email: [email protected])

Abstract

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We present a new algorithm for constructing a Chevalley basis for any Chevalley Lie algebra over a finite field. This is a necessary component for some constructive recognition algorithms of exceptional quasisimple groups of Lie type. When applied to a simple Chevalley Lie algebra in characteristic p⩾5, our algorithm has complexity involving the seventh power of the Lie rank, which is likely to be close to best possible.

MSC classification

Secondary: 17B45: Lie algebras of linear algebraic groups 20G40: Linear algebraic groups over finite fields
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 15 , May 2012 , pp. 436 - 443
Copyright
© The Author(s) 2012

References

[1]Carter, R. W., Simple groups of Lie type (Wiley, 1972).Google Scholar
[2]Celler, F. and Leedham-Green, C. R., ‘Calculating the order of an invertible matrix’, Groups and computation II (American Mathematical Society, providence, RI, 1997) 55–60.CrossRefGoogle Scholar
[3]Cohen, A. M. and Murray, S. H., ‘An algorithm for Lang’s theorem’, J. Algebra 322 (2009) 675–702.CrossRefGoogle Scholar
[4]Cohen, A. M. and Roozemond, D., ‘Computing Chevalley bases in small characteristics’, J. Algebra 322 (2009) 703–721.CrossRefGoogle Scholar
[5]Kantor, W. M. and Magaard, K., ‘Black box exceptional groups of Lie type’, Trans. Amer. Math. Soc. (to appear).Google Scholar
[6]Roozemond, D., ‘Computing maximal split toral subalgebras of Lie algebras over fields of small characteristic’, J. Symbolic Comput. 50 (2013) 335–349.CrossRefGoogle Scholar
[7]Ryba, A. J. E., ‘Computer construction of split Cartan subalgebras’, J. Algebra 309 (2007) 455–483.Google Scholar