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Algebras of generalized quaternion type: biregular case

Karin Erdmann, Adam Hajduk, Adam Skowyrski

Abstract

This paper provides the next step towards classification of algebras of generalized quaternion type. Previously algebras with 2-regular Gabriel quiver were classified (a quiver is 2-regular if at each vertex, two arrows start and two arrows end). Here we classify the algebras where at each vertex, either one arrow starts and one arrow ends, or else two arrows start and two arrows end. Our main result shows that that any such algebra (up to socle equivalence) is either a weighted surface algebra, or a higher spherical algebra.

Algebras of generalized quaternion type: biregular case

Abstract

This paper provides the next step towards classification of algebras of generalized quaternion type. Previously algebras with 2-regular Gabriel quiver were classified (a quiver is 2-regular if at each vertex, two arrows start and two arrows end). Here we classify the algebras where at each vertex, either one arrow starts and one arrow ends, or else two arrows start and two arrows end. Our main result shows that that any such algebra (up to socle equivalence) is either a weighted surface algebra, or a higher spherical algebra.
Paper Structure (18 sections, 29 theorems, 103 equations)

This paper contains 18 sections, 29 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Basic notions
  3. Weighted surface algebras
  4. Algebras of generalized quaternion type
  5. General properties
  6. Minimal relations around $1$-vertices
  7. Summary
  8. Bases of projectives
  9. Minimal relations starting from a $2$-vertex.
  10. Radical quotients and generators.
  11. Bases.
  12. Proof of Main Theorem
  13. Construction of $\hat{\Lambda}$
  14. Bases of projective $\hat{\Lambda}$-modules
  15. Symmetrizing form for $\hat{\Lambda}$
  16. ...and 3 more sections

Key Result

Theorem 3.1

Let $A$ be an algebra with $2$-regular Gabriel quiver having at least three vertices. Then the following statements are equivalent.

Theorems & Definitions (46)

  • Theorem 3.1
  • Theorem 3.2
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 4.4
  • Lemma 4.5
  • Proposition 4.6
  • Theorem 4.7
  • Lemma 4.8
  • ...and 36 more