GRADED LIE-ALGEBRAS OF MAXIMAL CLASS

We study graded Lie algebras of maximal class over a field F of positi ve characteristic p. A. Shalev has constructed infinitely many pairwis e non-isomorphic insoluble algebras of this kind, thus showing that th ese algebras are more complicated than might be suggested by consideri ng only associated Lie algebras of p-groups of maximal class. Here we construct {\F\aleph N-0} pairwise non-isomorphic such algebras, and ma x{\F\, aleph(0)} soluble ones. Both numbers are shown to be best possi ble. We also exhibit classes of examples with a non-periodic structure . As in the case of groups, two-step centralizers play an important ro le.