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The orders of nonsingular derivations

Part of: General field theory Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  09 April 2009

Aner Shalev
Aner Shalev
Affiliation:
Institute of Mathematics The Hebrew University Jerusalem 91904 Israel e-mail: shalev@math.huji.ac.il
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Abstract

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Nonsingular derivations of modular Lie algebras which have finite multiplicative order play a role in the coclass theory for pro-p groups and Lie algebras. We study the orders of nonsingular derivations of finite-dimensional non-nilpotent Lie algebras of characteristic p > 0. The methods are essentially number-theoretic.

1991 Mathematics subject classification (Amer. Math. Soc): primary 17B50; secondary 12E20.

MSC classification

Secondary: 17B50: Modular Lie (super)algebras 12E20: Finite fields (field-theoretic aspects)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 67 , Issue 2 , October 1999 , pp. 254 - 260
Copyright
Copyright © Australian Mathematical Society 1999

References

[AF]Albert, A. A. and Frank, M. S., ‘Simple Lie algebras of characteristic p’, Rend. Torino 14 (1954/1955), 117–139.Google Scholar
[BKK]Benkart, G., Kostrikin, A. I. and Kuznetsov, M. I., ‘Finite-dimensional Lie algebras with a nonsingular derivation’, J. Algebra 171 (1995), 894–916.CrossRefGoogle Scholar
[CMN]Caranti, A., Mattarei, S. and Newman, M. F., ‘Graded Lie algebras of maximal class’, Trans. Amer. Math. Soc. 349 (1997), 4021–4051.CrossRefGoogle Scholar
[CN]Caranti, A. and Newman, M. F., ‘Graded Lie algebras of maximal class II’, preprint, 1998.CrossRefGoogle Scholar
[J1]Jacobson, N., ‘A note on automorphisms and derivations of Lie algebras’, Proc. Amer. Math. Soc. 6 (1955), 281–283.CrossRefGoogle Scholar
[J2]Jacobson, N., Lie algebras (Wiley-Interscience, New York, 1962).Google Scholar
[Kh]Khukhro, E. I., Nilpotent groups and their automorphisms (de Gruyter, Berlin, 1993).CrossRefGoogle Scholar
[KK]Kostrikin, A. I. and Kuznetsov, M. I., ‘Lie algebras with a nonsingular derivation’, in: Algebra and analysis (Kazan, 1994) (de Gruyter, Berlin, 1996) pp. 81–90.Google Scholar
[LGN]Leedham-Green, C. R. and Newman, M. F., ‘Space groups and groups of prime power order I’, Arch. Math. 35 (1980), 193–202.CrossRefGoogle Scholar
[LN]Lidl, R. and Niederreiter, H., Introduction to finite fields and their applications (Cambridge Univ. Press, Cambridge, 1986).Google Scholar
[Sh1]Shalev, A., ‘The structure of finite p-groups: effective proof of the coclass conjectures’, Invent. Math. 115 (1994), 315–345.CrossRefGoogle Scholar
[Sh2]Shalev, A., ‘Simple Lie algebras and Lie algebras of maximal class’, Arch. Math. 63 (1994), 297–301.CrossRefGoogle Scholar
[ShZ]Shalev, A. and Zelmanov, E. I., ‘Pro-p groups of finite coclass’, Math. Proc. Cambridge Philos. Soc. 111 (1992), 417–421.CrossRefGoogle Scholar
[St]Strade, H., ‘The classification of the simple modular Lie algebras. VI. Solving the final case’, Trans. Amer. Math. Soc. 350 (1998), 2553–2628.CrossRefGoogle Scholar