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.
1
Università degli Studi di Milano-Bicocca, Milano, Italy
2
Charlotte Scott Centre for Algebra, University of Lincoln, United Kingdom
Marina Avitabile; Sandro Mattarei. The Earliest Diamond of Finite Type in Nottingham Algebras. Journal of Lie Theory, Tome 32 (2022) no. 3, pp. 771-796. http://geodesic.mathdoc.fr/item/JOLT_2022_32_3_a7/
@article{JOLT_2022_32_3_a7,
author = {Marina Avitabile and Sandro Mattarei},
title = {The {Earliest} {Diamond} of {Finite} {Type} in {Nottingham} {Algebras}},
journal = {Journal of Lie Theory},
pages = {771--796},
year = {2022},
volume = {32},
number = {3},
url = {http://geodesic.mathdoc.fr/item/JOLT_2022_32_3_a7/}
}
TY - JOUR
AU - Marina Avitabile
AU - Sandro Mattarei
TI - The Earliest Diamond of Finite Type in Nottingham Algebras
JO - Journal of Lie Theory
PY - 2022
SP - 771
EP - 796
VL - 32
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UR - http://geodesic.mathdoc.fr/item/JOLT_2022_32_3_a7/
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