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| 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.

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We investigate a class of infinite-dimensional, modular, graded Lie algebra in which the homogeneous components have dimension at most two. A subclass of these algebras can be obtained via a twisted loop algebra construction from certain finite-dimensional, simple Lie algebras of Albert-Frank type.Another subclass of these algebras is strictly related to certain graded Lie algebras of maximal class, and exhibits a wide range of behaviours.1991 Mathematics subject classification (Amer. Math. Soc): primary 17B50, 17B70, 17B65, 17B68, 17B56.

Text. We prove congruences, modulo a power of a prime p, for certain finite sums involving central binomial coefficients ((2k)(k)), partly motivated by analogies with the well-known power series for (arcsin z)(2) and (arcsin z)(4). The right-hand sides of those congruences involve values of the finite polylogarithms L-d(x) = Sigma(p-1)(k=1) x(k)/k(d). Exploiting the available functional equations for the latter we compute those values, modulo the required powers of p, in terms of familiar quantities such as Fermat quotients and Bernoulli numbers. Video. For a video summary of this paper, please click here or visit http://www.youtube.com/warch?v-W54Ad0YFr8A. (C) 2012 Elsevier Inc. All rights reserved

A thin Lie algebra is a Lie algebra graded over the positive integers satisfying a certain narrowness condition. We describe several cyclic grading of the modular Hamiltonian Lie algebras H(2: n; (02) (of dimension one less than a power of p) from which we construct infinite-dimensional thin Lie algebras. In the process we provide an explicit identification of H(2: n;0)2) with a Block algebra. We also compute its second cohomology group and its derivation algebra (in arbitrary prime characteristic).2000 Mathematics subject classification: primary 17B50; secondary 17B70, 17B56, 17B65.