We study graded Lie algebras of maximal class over a field 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 {\F\aleph N-0} pairwise non-isomorphic such algebras, and max{\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.
Caranti, A., Mattarei, S., Newman, M. (1997). Graded Lie algebras of maximal class. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 349(10), 4021-4051 [10.1090/S0002-9947-97-02005-9].
Graded Lie algebras of maximal class
Mattarei, S;
1997
Abstract
We study graded Lie algebras of maximal class over a field 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 {\F\aleph N-0} pairwise non-isomorphic such algebras, and max{\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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