Graded Lie algebras of maximal class, III

Abstract

We describe the isomorphism classes of infinite-dimensional N-graded Lie algebras of maximal class generated by their first homogeneous component over fields of characteristic two. This complements the analogous work by Caranti and Newman in the odd characteristic case.  2004 Elsevier Inc. All rights reserved.

Thin Lie algebras with diamonds of finite and infinite type

“…An impression that there might be few thin Lie algebras after all is, however, dissipated by one of the results of [CM99], which sets up a correspondence between a certain subclass of infinitedimensional thin Lie algebras with second diamond in degree 2q −1 (namely, those with all diamonds of infinite type, see the next paragraph) and a subclass of the infinite-dimensional graded Lie algebras of maximal class introduced in [CMN97] (namely, those with exactly two distinct two-step centralizers). Although infinite-dimensional graded Lie algebras of maximal class do admit a classification [CN00,Jur05], the subclass under consideration is very large, as it contains uncountably many isomorphism types. Their abundance is also illustrated by the fact that most of them, in an appropriate sense, are not determined by any of their finite-dimensional quotients.…”

A family of simple Lie algebras in characteristic two

“…For our purposes, we can rely on the results listed in Propositions 3.3-3.6 in [9], where p is the characteristic of the field (in our case p = 2). Since an algebra is inflated if and only if all constituent lengths are multiples of p, neither the Bi-Zassenhaus loop algebras nor the Albert-Frank-Shalev algebras are inflated: actually, they are the only non-inflated ones [19]. Given any algebra of maximal class L, its inflation L ↑ can be obtained by multiplying all its constituents lengths by p: then, if we denote by q ↑ = p h+1 the parameter of the inflated algebra, we have…”

“…The family of Bi-Zassenhaus loop algebras plays an important role in the classification [19] of infinite-dimensional modular N-graded Lie algebras of maximal class. In the odd characteristic case, every infinite-dimensional modular N-graded Lie algebra of maximal class (generated by its first homogeneous component) can be built starting from an Albert-Frank-Shalev algebra via some suitable constructions, as shown by Caranti and Newman in [9].…”

Thin loop algebras of Albert–Zassenhaus algebras

“…Those were first considered in [Sha94] to provide Lie algebra counterexamples to analogues of the coclass conjectures of pro-p-group theory (see [LGM02], for example), and were later studied and classified in [CMN97,CN00,Jur05]. In fact, graded Lie algebras of maximal class satisfy the definition of thin Lie algebras given above, but have no diamonds except the first one.…”