The title matches that of a series of papers by various authors beginning in 1997, whose goal was the study and classification of such algebras over fields of positive characteristic. The original motivation came from group theory: the Leedham-Green and Newman coclass conjectures on pro-p groups from 1980 had all become theorems relatively recently, and subsequent results of Shalev and Zelmanov had raised interest in what one could say about Lie algebras of finite coclass. In positive characteristic, the simplest case of coclass one (i.e., 'Lie algebras of maximal class', also called 'filiform' in some quarters) appeared challenging even under the strong assumptions of those Lie algebras being infinite-dimensional and graded over the positive integers. I will review motivations and results of those studies, including some classifications obtained by Caranti, Newman, Vaughan-Lee. Then I will describe some generalizations recently established with three of my former PhD students.
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