Thin loop algebras of Albert–Zassenhaus algebras

Abstract

Thin Lie algebras are Lie algebras over a field, graded over the positive integers and satisfying a certain narrowness condition. In particular, all homogeneous components have dimension one or two, and are called diamonds in the latter case. The first diamond is the component of degree one, and the second diamond can only occur in degrees 3, 5, q or 2q − 1, where q is a power of the characteristic of the underlying field. Here we consider several classes of thin Lie algebras with second diamond in degree q. In particular, we identify the Lie algebras in one of these classes with suitable loop algebras of certain Albert-Zassenhaus Lie algebras. We also apply a deformation technique to recover other thin Lie algebras previously produced as loop algebras of certain graded Hamiltonian Lie algebras.

A sandwich in thin lie algebras

“…We discuss how far Theorem 11 may be extended to include thin Lie algebras of characteristic five, with second diamond L 5 and an additional assumption. Countably many thin Lie algebras with second diamond L q were constructed in [4], for any power q of the odd characteristic. In particular, when q = 5, this gives us a countable family of thin Lie algebras with second diamond L 5 .…”

“…Its subalgebra L = k>0 Sk ⊗ t k , where k = k + N Z, is naturally graded over the positive integers, and is called a loop algebra in this context. In certain cases, one needs a slightly more general construction involving also a derivation of S (such as that of [4,Definition 2.1]), but that is inconsequential for our present observation.…”

“…In contrast, the values q and 2q − 1 for k occur for two broad classes of thin Lie algebras, of which various families were built from certain non-classical finite-dimensional simple modular Lie algebras, but also to other thin Lie algebras obtained from graded Lie algebras of maximal class through various constructions. General discussions of those two classes of thin Lie algebras may be found in [4] and [11], respectively.…”

A sandwich in thin Lie algebras

“…We discuss how far Theorem 11 may be extended to include thin Lie algebras of characteristic five, with second diamond L 5 and an additional assumption. Countably many thin Lie algebras with second diamond L q were constructed in [AM07], for any power q of the odd characteristic. In particular, when q = 5 this gives us a countable family of thin Lie algebras with second diamond L 5 .…”

“…In contrast, the values q and 2q − 1 for k occur for two broad classes of thin Lie algebras, of which many were built from certain non-classical finite-dimensional simple modular Lie algebras, and also to other thin Lie algebras obtained from graded Lie algebras of maximal class through various constructions. General discussions of those two classes of thin Lie algebras can be found in [AM07] and [CM05], respectively.…”