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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: 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
MR Author ID: 45160
ORCID: 0000-0002-5746-9294

S. Mattarei
Affiliation: Dipartimento di Matematica ed Applicazioni, Università degli Studi di Padova, via Belzoni 7, I-35131 Padova, Italy

M. F. Newman
Affiliation: School of Mathematical Sciences, Australian National University, Canberra, ACT 0200, Australia

Keywords: Graded Lie algebras of maximal class, p-groups of maximal class, pro-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