School of Mathematics & Physics, University of Lincoln
A collaboration across countries and over several years has taken place between Sandro Mattarei and two of his former PhD students, Claudio Scarbolo and Valentina Iusa, resulting in a preprint which was recently submitted for publication: “Graded Lie algebras of maximal class of type p“. Valentina was the first Algebra PhD student of the Charlotte Scott Algebra Centre and worked under the joint supervision of Sandro and Evgeny Khukhro.
The preprint concerns infinite-dimensional Lie algebras, over a field of prime characteristic p, which are graded over the positive integers and have maximal class in the sense that the kth Lie power of L has codimension k in L, for all k>1. The type p qualifier refers to the fact that the focus is on those generated by an element of degree p, together with an element of degree 1. (The actual natural definition is a bit more restrictive.) When the prime p equals 2, such Lie algebras were completely classified by Caranti and Vaughan-Lee in 2003. An extension to odd primes p presents many challenges which were addressed in Claudio’s PhD thesis (Trento, 2014). Claudio achieved a classification but at the expense of quite intricate calculations. Turning some of those calculations into more readable arguments was done by Valentina in a portion of her PhD thesis (Lincoln, 2019), and required producing new ideas of wider applicability. Finally, further simplification work, and addition of new results, were done by the authors through an online collaboration over several months.
The 40-page preprint includes several main parts of the classification proof, but not all, and a separate paper is planned for completing its exposition. A generalization to type n, meaning still in characteristic p but generated by an element of degree 1 and one of arbitrary degree n, is a bigger challenge. Only the special cases n=1, 2, p have been settled. Some preliminary steps in the general case were taken by Sandro’s former student Simone Ugolini in his PhD thesis (Trento, 2010), and we hope that those results will also eventually find their way to publication. We also believe that some of the techniques developed in our recent preprint will allow progress on the general case of type n.