Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Extended Admissible Dissections of Marked Surfaces and Piano Algebras

Marina Godinho, Dave Murphy

Abstract

We introduce the notion of extended admissible dissections of a marked surface, building upon the notion of an admissible dissection of a marked surface by Amiot--Plamondon--Schroll. For each extended admissible dissection we construct a differential graded algebra, called a piano algebra, which may be viewed in some sense as a differential graded analogue of a gentle algebra. We show that for a marked disc without punctures, a piano algebra is quasi-isomorphic to the graded endomorphism ring of a classical generator of the Paquette--Yıldırım completion of the discrete cluster category of Dynkin type $A_{\infty}$, labelled $\overline{\mathcal{C}}_n$. We use previous results of the authors to show that there exists an additive equivalence between $\overline{\mathcal{C}}_n$ and the perfect derived category of a specific piano algebra, that sends triangles with two indecomposable terms to triangles with two indecomposable terms. We use this equivalence to prove that any two piano algebras coming from homeomorphic marked discs are derived equivalent.

Extended Admissible Dissections of Marked Surfaces and Piano Algebras

Abstract

We introduce the notion of extended admissible dissections of a marked surface, building upon the notion of an admissible dissection of a marked surface by Amiot--Plamondon--Schroll. For each extended admissible dissection we construct a differential graded algebra, called a piano algebra, which may be viewed in some sense as a differential graded analogue of a gentle algebra. We show that for a marked disc without punctures, a piano algebra is quasi-isomorphic to the graded endomorphism ring of a classical generator of the Paquette--Yıldırım completion of the discrete cluster category of Dynkin type , labelled . We use previous results of the authors to show that there exists an additive equivalence between and the perfect derived category of a specific piano algebra, that sends triangles with two indecomposable terms to triangles with two indecomposable terms. We use this equivalence to prove that any two piano algebras coming from homeomorphic marked discs are derived equivalent.
Paper Structure (15 sections, 45 theorems, 77 equations, 8 figures)

This paper contains 15 sections, 45 theorems, 77 equations, 8 figures.

Table of Contents

  1. Discrete Cluster Categories
  2. Discrete Cluster Categories of Dynkin Type Ainf
  3. Classical Generators of Cn
  4. Dissections of Marked Surfaces
  5. Admissible Dissections of Marked Surfaces
  6. The Locally Gentle Algebra of an Admissible Dissection
  7. Extended Admissible Dissections
  8. Piano Algebras
  9. Classical Generators and Admissible Dissections of a Disc
  10. Graded Endomorphism Ring of Generators in Cn
  11. Graded Endomorphism Rings
  12. Piano Algebras and Classical Generators
  13. Fan Generators of Cn
  14. Derived Equivalence of Piano Algebras from Discs
  15. Derived Equivalences of Piano Algebras

Key Result

Theorem 1

There exists a bijection between the homeomorphism classes of extended admissible dissections of $(D^2,M_n,\emptyset)$, and the equivalence classes of limit generators of $\overline{\EuScript{C}}_{n}$.

Figures (8)

  • Figure 1: An admissible subset $\mathscr{M}$ of $S^1$. The marked points in $\mathscr{M}$ converge to the accumulation points represented as small circles, and each marked point $x$ has both a predecessor and a successor, labelled $x^-$ and $x^+$ respectively.
  • Figure 2: Four arcs of $\overline{\mathscr{M}}_{}$, where $\ell_X$ is a short arc, $\ell_Y$ is a long arc, $\ell_W$ is a limit arc, and $\ell_Z$ is a double limit arc.
  • Figure 3: The arcs corresponding to triangles coming from non-split extensions between indecomposable objects.
  • Figure 4: The objects $G$ and $H$ are an example of two generators such that $H$ and $G$ are in the same equivalence class.
  • Figure 5: An admissible $\textcolor{red}{\circ}$-dissection and an admissible $\textcolor{green}{\bullet}$-dissection of the marked surface $(D^2,M,P)$.
  • ...and 3 more figures

Theorems & Definitions (97)

  • Theorem 1: \ref{['Prop: Bijection of ext admissible dissections']}
  • Theorem 2: \ref{['Thm:path algebras']} and \ref{['Lem: Lambda 0 is gentle']}
  • Theorem 3: \ref{['Thm: Derived Equiv']}
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Proposition 1.4
  • Lemma 1.5
  • proof
  • Corollary 1.6
  • ...and 87 more