Graded Lie Algebras of Maximal Class

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.