Some thin Lie algebras related to Albert-Frank algebras and algebras of maximal class

Abstract

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.

Gradings of non-graded Hamiltonian Lie algebras

“…Nevertheless, Shalev's algebras occupy a unique place in the description of the graded Lie algebras of maximal class 402 A. Caranti and S. Mattarei [4] follow for more details.) The arguments of [23] have been extended in [4,20] to show that the second diamond in an infinite-dimensional thin Lie algebra (or one of finite dimension large enough) can only occur in degree 3, 5, q or 2q -1, for some power q of the characteristic p of the underlying field. It follows from [23] that there are, up to isomorphism and with the possible exception of very small characteristics, one or two (depending on the ground field) infinite-dimensional thin Lie algebras with second diamond in degree 3 and no diamond in degree 4, and one with second diamond in degree 5.…”

“…Several of these subcases have been investigated in various papers, having in mind a classification of all infinite-dimensional thin Lie algebras as a distant goal. We refer to the paper [21], and to the references mentioned there, for a discussion of thin Lie algebras with second diamond in degree q, and to [20,5] for those with second diamond in degree 1q -1. Here we restrict ourselves to some general and informal comments on the type of results which have been proved so far.…”

“…In particular, infinite-dimensional thin Lie algebras with second diamond in degree q have been constructed as loop algebras of Zassenhaus algebras (which have dimension [5] Gradings of non-graded Hamiltonian Lie algebras 403 a power of p, see [16,17,18]), and Hamiltonian algebras of the types H(2 : n;co 0 ) = H(2 : n) (the graded simple Hamiltonian algebras, of dimension two less than a power of p, see [3]) and H (2 : n; u>\) (which are Albert-Zassenhaus algebras and have dimension a power of p, see [6]). A preliminary version of the present paper, which predated and inspired some of the other papers cited here, had as its main goal the construction of some infinitedimensional thin Lie algebras with second diamond in degree 2q -1 as loop algebras of Hamiltonian algebras //(2 : n; co 2 ), which have dimension one less than a power of p. (In fact, a construction for those thin Lie algebras had already been given in [20], but as loop algebras of certain finite-dimensional Lie algebras defined 'ad hoc'.) The paper has somehow expanded after we have realised that some of our result may be of interest independently of their application to thin Lie algebras.…”

“…We describe that in Subsection 5.2 as a loop algebra of H(2: n; co 2 ) with respect to a suitable grading. In [20] we also constructed thin Lie algebras with second diamond in degree 2q -1 and all diamonds of finite types, as loop algebras of certain Block algebras. The Block algebras used in [20] are actually isomorphic with algebras of Albert and Frank (being simple Block algebras with G = G o , see Section 3, and according to known results, for example [13,Lemma 1.8.3]), but were presented there in a different basis.…”

“…In [20] we also constructed thin Lie algebras with second diamond in degree 2q -1 and all diamonds of finite types, as loop algebras of certain Block algebras. The Block algebras used in [20] are actually isomorphic with algebras of Albert and Frank (being simple Block algebras with G = G o , see Section 3, and according to known results, for example [13,Lemma 1.8.3]), but were presented there in a different basis. As we have mentioned earlier, the original motivation for the present paper was finding an explicit identification of those algebras with Hamiltonian algebras H{2 : n; <o 2 ) and describing a corresponding cyclic grading [7] Gradings of non-graded Hamiltonian Lie algebras 405 of the latter.…”