We prove several structural results on Nottingham algebras, a class of infinitedimensional, 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.

Avitabile, M., Mattarei, S. (2022). The Earliest Diamond of Finite Type in Nottingham Algebras. JOURNAL OF LIE THEORY, 32(3), 771-796 [10.48550/arXiv.2106.14796].

The Earliest Diamond of Finite Type in Nottingham Algebras

Avitabile M.
;
Mattarei S.
2022

Abstract

We prove several structural results on Nottingham algebras, a class of infinitedimensional, 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.
Articolo in rivista - Articolo scientifico
graded Lie algebra; Modular Lie algebra; thin Lie algebra;
English
771
796
26
Avitabile, M., Mattarei, S. (2022). The Earliest Diamond of Finite Type in Nottingham Algebras. JOURNAL OF LIE THEORY, 32(3), 771-796 [10.48550/arXiv.2106.14796].
Avitabile, M; Mattarei, S
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/393439
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