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.
The group generated by the round functions of a block ciphers is a widely investigated problem. We identify a large class of block ciphers for which such group is easily guaranteed to be primitive. Our class includes the AES and the SERPENT.
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.
The graded Lie algebra L associated to the Nottingham group is a loop algebrâ of the Witt algebra W . The universal covering W of W has one-dimensional 1 1 1 centre, so that the corresponding loop algebra M of W has an infinite-dimen-H. Neumann that L is not finitely presented. However, we are able to show that M itself is finitely presented.We work more generally with the Zassenhaus algebras W . In the group context, n examples of finitely presented groups whose centre is not finitely generated were given
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