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Classification of Nottingham algebras

M. Avitabile, A. Caranti, S. Mattarei

Abstract

The graded Lie algebra associated with the Nottingham group over a field of prime characteristic serves as a fundamental example of Nottingham algebras, a class of infinite-dimensional, positively graded thin algebras. This paper completes the classification of Nottingham algebras initiated in earlier papers, proving both existence and uniqueness results that determine all such algebras up to isomorphism.

Classification of Nottingham algebras

Abstract

The graded Lie algebra associated with the Nottingham group over a field of prime characteristic serves as a fundamental example of Nottingham algebras, a class of infinite-dimensional, positively graded thin algebras. This paper completes the classification of Nottingham algebras initiated in earlier papers, proving both existence and uniqueness results that determine all such algebras up to isomorphism.
Paper Structure (12 sections, 18 theorems, 52 equations)

This paper contains 12 sections, 18 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Nottingham algebras and their diamonds
  3. The double grading of a Nottingham algebra
  4. Regular Nottingham algebras
  5. The other families of Nottingham algebras
  6. The algebras $\mathcal{L}_{1,q}$ and $\mathcal{L}_{0,q}$
  7. The Nottingham deflations $\mathcal{N}(q,r)$
  8. The algebras $\mathcal{T}_{q,1}(M)$ and $\mathcal{T}_{q,2}(M)$ obtained from algebras of maximal class
  9. Irregular Nottingham algebras
  10. The class $\mathcal{T}_{q,2}$ of Nottingham algebras.
  11. A membership criterion for $\mathcal{T}_{q,2}$
  12. General calculations in a Nottingham algebra

Key Result

Theorem 2.5

Let $L$ be a Nottingham algebra with second diamond $L_q$, and standard generators $x$ and $y$. Let $L_m$ be a (possibly fake) diamond of $L$, with $m\ge q$.

Theorems & Definitions (37)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.5: Theorem 3.1 in AviMat:earliest
  • Theorem 4.1: Theorem 7 in AviMat:diamond_distances
  • Theorem 6.1
  • Theorem 6.2: Theorem 5.7 in Young:thesis
  • Theorem 6.3
  • Definition 7.1
  • Proposition 7.2: Theorem 3.3 in Young:thesis
  • ...and 27 more