Journal of Lie Theory 32 (2022), No. 3, 771--796
Copyright Heldermann Verlag 2022
The Earliest Diamond of Finite Type in Nottingham Algebras
UniversitÓ degli Studi di Milano-Bicocca, Milano, Italy
Charlotte Scott Centre for Algebra, University of Lincoln, United Kingdom
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.
Keywords: Modular Lie algebra, graded Lie algebra, thin Lie algebra.
MSC: 17B50; 17B70, 17B65.
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