School of Mathematics & Physics, University of Lincoln
A paper by Marina Avitabile and Sandro Mattarei, The earliest diamond of finite type in Nottingham algebras, has been accepted for publication in Journal of Lie Theory. (See https://arxiv.org/abs/2106.14796 for a preprint version. The photograph is a portrait of Sophus Lie, 1896.)
Abstract: We prove several structural results on Nottingham algebras, a class of infinite-dimensional, modular, graded Lie algebras, which includes the graded Lie algebra associated to the Nottingham group with respect to its lower central series. Homogeneous components of a Nottingham algebra have dimension one or two, and in the latter case they are called diamonds. The first diamond occurs in degree 1, and the second occurs in degree q, a power of the characteristic. Each diamond past the second is assigned a type, which either belongs to the underlying field or is ∞. Nottingham algebras with a variety of diamond patterns are known. In particular, some have diamonds of both finite and infinite type. We prove that each of those known examples is uniquely determined by a certain finite-dimensional quotient. Finally, we determine how many diamonds of type ∞ may precede the earliest diamond of finite type in an arbitrary Nottingham algebra.