Graded Lie Algebras of Maximal Class

Caranti, A.; Mattarei, S.; Newman, M.

doi:10.1090/S0002-9947-97-02005-9

Graded Lie Algebras of Maximal Class
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by A. Caranti, S. Mattarei and M. F. Newman PDF

Trans. Amer. Math. Soc. 349 (1997), 4021-4051 Request permission

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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