The earliest diamond of finite type in Nottingham algebras

Avitabile, Marina and Mattarei, Sandro (2022) The earliest diamond of finite type in Nottingham algebras. Journal of Lie Theory . ISSN 0949-5932

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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 infinity.

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 infinity may precede the earliest diamond of finite type in an arbitrary Nottingham algebra.

Keywords:Modular Lie algebra, graded Lie algebra, thin Lie algebra
Subjects:G Mathematical and Computer Sciences > G110 Pure Mathematics
Divisions:College of Science > School of Mathematics and Physics
ID Code:49518
Deposited On:25 May 2022 10:36

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