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Wild Local Structures of Automorphic Lie Algebras

Drew Duffield, Vincent Knibbeler, Sara Lombardo

Abstract

We study automorphic Lie algebras using a family of evaluation maps parametrised by the representations of the associative algebra of functions. This provides a descending chain of ideals for the automorphic Lie algebra which is used to prove that it is of wild representation type. We show that the associated quotients of the automorphic Lie algebra are isomorphic to twisted truncated polynomial current algebras. When a simple Lie algebra is used in the construction, this allows us to describe the local Lie structure of the automorphic Lie algebra in terms of affine Kac-Moody algebras.

Wild Local Structures of Automorphic Lie Algebras

Abstract

We study automorphic Lie algebras using a family of evaluation maps parametrised by the representations of the associative algebra of functions. This provides a descending chain of ideals for the automorphic Lie algebra which is used to prove that it is of wild representation type. We show that the associated quotients of the automorphic Lie algebra are isomorphic to twisted truncated polynomial current algebras. When a simple Lie algebra is used in the construction, this allows us to describe the local Lie structure of the automorphic Lie algebra in terms of affine Kac-Moody algebras.

Paper Structure

This paper contains 17 sections, 17 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Point-Evaluation Representations
  4. Filtrations of Representations
  5. Wild Representation Type
  6. The Finitely Generated Representation Theory of $\mathcal{O}_\mathbb{X}$ when $\mathbb{Y}$ is Genus Zero
  7. The indecomposable representations
  8. The morphisms of $\mathop{\mathrm{fin}}\nolimits \mathcal{O}_\mathbb{X}$
  9. Representations of $\mathcal{O}_\mathbb{X}$ Arising from Jet Maps
  10. Ideals and Quotients of Automorphic Lie Algebras
  11. Ideals Arising from Jet Representations
  12. The Wildness of Automorphic Lie Algebras
  13. Twisted Truncated Current Algebras as Quotients of Automorphic Lie Algebras
  14. The Local Structure of Automorphic Lie Algebras
  15. Further Directions
  16. ...and 2 more sections

Key Result

Proposition 2.5

An additive, full, exact subcategory $\mathcal{C} \subseteq \mathop{\mathrm{fin}}\nolimits \mathfrak{A}$ is wild if and only if for any finite-dimensional associative algebra $\Lambda$, there exists an exact $K$-linear functor $F\colon \mathop{\mathrm{fin}}\nolimits \Lambda \rightarrow \mathcal{C}$

Theorems & Definitions (43)

  • Definition 2.1: neher2012irreducible
  • Definition 2.2: neher2012irreducible
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5: BlueBookIII, XIX.1.11
  • Theorem 2.6: makedonskii2013on, Theorem 3
  • Definition 2.7
  • Remark 2.8
  • Remark 2.9
  • Remark 2.10
  • ...and 33 more