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Hybrid algebras

Karin Erdmann, Andrzej Skowroński

Abstract

We introduce a new class of symmetric algebras, which we call hybrid algebras. This class contains on one extreme Brauer graph algebras, and on the other extreme general weighted surface algebras. We show that hybrid algebras are precisely the blocks of idempotent algebras of weighted surface algebras, up to socle deformations. More generally, for tame symmetric algebras whose Gabriel quiver is 2-regular, we show that the tree class of an Auslander-Reiten component is Dynkin or Euclidean or one of the infinite tress $A_{\infty}$, $A_{\infty}^{\infty}$, or $D_{\infty}$.

Hybrid algebras

Abstract

We introduce a new class of symmetric algebras, which we call hybrid algebras. This class contains on one extreme Brauer graph algebras, and on the other extreme general weighted surface algebras. We show that hybrid algebras are precisely the blocks of idempotent algebras of weighted surface algebras, up to socle deformations. More generally, for tame symmetric algebras whose Gabriel quiver is 2-regular, we show that the tree class of an Auslander-Reiten component is Dynkin or Euclidean or one of the infinite tress , , or .

Paper Structure

This paper contains 33 sections, 32 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and regular hybrid algebras
  3. The setup
  4. Notation
  5. Regular hybrid algebras
  6. Idempotent algebras of WSA's
  7. General hybrid algebras
  8. The exceptions in relations (2) and (2')
  9. The subquiver around a critical arrow
  10. Subquivers around a virtual arrow
  11. Consistency, bases and dimensions
  12. Some hybrid algebras with at most three simple modules
  13. Local algebras
  14. Hybrid algebras with two simple modules
  15. Algebras with triangular quiver
  16. ...and 18 more sections

Key Result

Theorem 1.1

(i) Assume $\Lambda$ is a weighted surface algebra and $e$ is an idempotent of $\Lambda$, then every block component of $e\Lambda e$ is a hybrid algebra (up to socle equivalence). (ii) Assume $H$ is a hybrid algebra. Then there is a weighted surface algebra $\Lambda$ and an idempotent $e$ of $\Lambd

Theorems & Definitions (65)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Example 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 55 more