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Generalized weighted surface algebras

Andrzej Skowroński, Adam Skowyrski

Abstract

The weighted triangulation algebras associated to triangulation quivers and their socle deformations were recently introduced and studied in [15]-[20] and [2]. These algebras, based on surface triangulations and originated from the theory of cluster algebras, were also proved to be (with some minor exceptions) finite-dimensional tame symmetric and periodic algebras of period 4. In this paper, we introduce a new concept of a generalized triangulation quiver, extending the notion of a triangulation quiver. Inparticular, it is also shown that the generalized triangulation quivers can be constructed from triangulations of orientable surfaces with marked self-foldedtriangles. Moreover, motivated by the recent results of [28], we define and investigate so called weighted generalized triangulation algebras associated to generalized triangulation quivers, which naturally arise from mutations of weighted triangulation algebras.

Generalized weighted surface algebras

Abstract

The weighted triangulation algebras associated to triangulation quivers and their socle deformations were recently introduced and studied in [15]-[20] and [2]. These algebras, based on surface triangulations and originated from the theory of cluster algebras, were also proved to be (with some minor exceptions) finite-dimensional tame symmetric and periodic algebras of period 4. In this paper, we introduce a new concept of a generalized triangulation quiver, extending the notion of a triangulation quiver. Inparticular, it is also shown that the generalized triangulation quivers can be constructed from triangulations of orientable surfaces with marked self-foldedtriangles. Moreover, motivated by the recent results of [28], we define and investigate so called weighted generalized triangulation algebras associated to generalized triangulation quivers, which naturally arise from mutations of weighted triangulation algebras.

Paper Structure

This paper contains 8 sections, 23 theorems, 97 equations.

Table of Contents

  1. Introduction and the main results
  2. Derived equivalences of algebras
  3. Generalized triangulation quivers
  4. Weighted generalized triangulation algebras
  5. Basis and dimension
  6. Weighted triangulation algebras
  7. Proofs of Theorems 1 and 2
  8. Generalized triangulation quivers from surfaces

Key Result

Theorem 2.1

Algebras $A$ and $B$ are derived equivalent if and only if there exists a tilting complex $T$ in $K^b(P_A)$ such that $\operatorname{End}_{K^b(P_A)}\cong B$.

Theorems & Definitions (47)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Proposition 2.5
  • proof
  • Definition 3.1
  • Proposition 3.2
  • Example 3.3
  • Definition 4.1
  • ...and 37 more