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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Graded Lie Algebras of Maximal Class


Authors: A. Caranti, S. Mattarei and M. F. Newman
Journal: Trans. Amer. Math. Soc. 349 (1997), 4021-4051
MSC (1991): Primary 17B70, 17B65, 17B05, 17B30, 17B40, 20D15, 20F40
MathSciNet review: 1443190
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Abstract | References | Similar Articles | Additional Information

Abstract: We study graded Lie algebras of maximal class over a field $ \mathbf {F}$ of positive characteristic $p$. A. Shalev has constructed infinitely many pairwise non-isomorphic insoluble algebras of this kind, thus showing that these algebras are more complicated than might be suggested by considering only associated Lie algebras of p-groups of maximal class. Here we construct $| \mathbf {F}|^{\aleph _{0}}$ pairwise non-isomorphic such algebras, and $\max \{| \mathbf {F}|, \aleph _{0} \}$ soluble ones. Both numbers are shown to be best possible. We also exhibit classes of examples with a non-periodic structure. As in the case of groups, two-step centralizers play an important role.


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Additional Information

A. Caranti
Affiliation: Dipartimento di Matematica, Università degli Studi di Trento, via Sommarive 14, I-38050 Povo (Trento), Italy
Email: caranti@science.unitn.it

S. Mattarei
Affiliation: Dipartimento di Matematica ed Applicazioni, Università degli Studi di Padova, via Belzoni 7, I-35131 Padova, Italy
Email: mattarei@pdmat1.math.unipd.it

M. F. Newman
Affiliation: School of Mathematical Sciences, Australian National University, Canberra, ACT 0200, Australia
Email: newman@maths.anu.edu.au

DOI: http://dx.doi.org/10.1090/S0002-9947-97-02005-9
PII: S 0002-9947(97)02005-9
Keywords: Graded Lie algebras of maximal class, {\em p}\/-groups of maximal class, pro-{\em p}\/-groups of finite coclass, nilpotent Lie algebras
Received by editor(s): March 1, 1996
Additional Notes: The first two authors are members of CNR–GNSAGA, Italy, and acknowledge support of MURST, Italy. The third author acknowledges support from CNR-GNSAGA, Italy, and the University of Trento, Italy.
Article copyright: © Copyright 1997 American Mathematical Society