TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 349, Number 10, October 1997, Pages 4021–4051 S 0002-9947(97)02005-9 GRADED LIE ALGEBRAS OF MAXIMAL CLASS A. CARANTI, S. MATTAREI, AND M. F. NEWMAN Abstract. We study graded Lie algebras of maximal class over a field F of positive characteristic p. A. Shalev has constructed infinitely many pair- wise 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 pairwise non-isomorphic such algebras, and max{|F|, ℵ 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. 1. Introduction In a recent paper Shalev [Sh2] has shown that graded Lie algebras of maximal class are more complicated than might be suggested by considering only associated Lie algebras of p -groups with maximal class. A nilpotent graded Lie algebra L has maximal class if it has class c and dimension c + 1. Thus, L = L 1 ⊕ L 2 ⊕···⊕ L c with dim L 1 = 2 and, for 2 ≤ i ≤ c, dim L i = 1 and [L i−1 L 1 ]= L i . Shalev shows that over fields with positive characteristic there are such algebras with arbitrarily large soluble length. In this paper we exhibit many more examples, and prove some structure theorems which show that for every field F with positive characteristic there are max(|F|, ℵ 0 ) isomorphism types of nilpotent graded Lie algebras of maximal class over F. As usual we include under the heading ‘maximal class’ algebras L graded by the positive integers; that is, L = L 1 ⊕ L 2 ⊕···⊕ L i ⊕ ... with dim L 1 = 2 and, for 2 ≤ i, dim L i = 1 and [L i−1 L 1 ]= L i . These algebras can be viewed as (projective) limits of nilpotent graded Lie algebras of maximal class. They are just infinite-dimensional in the sense that, correctly interpreted, all their proper quotients are finite-dimensional. Shalev ([Sh2], Theorem 1) showed that there are insoluble algebras of this kind. We show that over every field F with Received by the editors March 1, 1996. 1991 Mathematics Subject Classification. Primary 17B70, 17B65, 17B05, 17B30, 17B40, 20D15, 20F40. Key words and phrases. Graded Lie algebras of maximal class, p-groups of maximal class, pro-p-groups of finite coclass, nilpotent Lie algebras. The first two authors are members of CNR–GNSAGA, Italy, and acknowledge support of MURST, Italy. The third author acknowledges support from CNR-GNSAGA, Italy, and the University of Trento, Italy. c 1997 American Mathematical Society 4021 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use