Graded Lie Algebras of Maximal Class

We study graded Lie algebras of maximal class over a field F F of positive characteristic p 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 | ℵ 0 | F|^{ ₀} pairwise non-isomorphic such algebras, and max { | F | , ℵ 0 } max \{| F|, 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.