Title:Thin subalgebras of Lie algebras of maximal class 

Abstract:  (Joint work with M. Avitabile, A. Caranti, V. Monti, M. F. Newman and E. O’Brien).
Let L be an infinite dimensional Lie algebra which is graded over the positive integers and is generated by its first homogeneous component L_1. The algebra L is of maximal class if dim(L_1)=2 and dim(L_i)=1 for i larger than 1. The algebra L is thin if it is not of maximal class, dim(L_1)=2 and L_{i+1}=[x,L_1] for any nontrivial element x in L_i.

Suppose that E is a quadratic extension of a field F and that M is a Lie algebra of maximal class over E. We consider the Lie algebra L generated over the field F by an F-subspace L_1 of M_1 having dimension 2 over F. We give necessary and sufficient conditions for the Lie algebra L to be a thin graded F-subalgebra of the F-algebra M. We show also that there are uncountably many such thin algebras that can be constructed by way of this “recipe”, attaining the maximum possible cardinality.

The authors started this project almost independently since 1999 and their partial results have been luckily and duly recorded by A. Caranti. Only recently we have been able to jointly develop thorough and concise results for this research.