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Jacobian Algebras of Species with Potentials and 2-Representation Finite Algebras

Christoffer Söderberg

Abstract

We study $2$-representation finite $\mathbb{K}$-algebras obtained from tensor products of tensor algebras of species. In earlier work we computed the higher preprojective algebra of said algebras to be given as Jacobian algebras of certain species with potential $(S, W)$, which are self-injective and finite dimensional. Truncating these Jacobian algebras yields a rich source of $2$-representation finite $\mathbb{K}$-algebras. Under suitable assumptions, we prove that the set of all cuts of $(S, W)$ is transitive under successive cut-mutations. Furthermore, we show that cuts and cut-mutation correspond to truncated Jacobian algebras and $2$-APR tilting, respectively. Consequently, under certain assumptions, all truncated Jacobian algebras are related to each other via $2$-APR tilting. We produce various new examples of $2$-representation finite $\mathbb{K}$-algebras.

Jacobian Algebras of Species with Potentials and 2-Representation Finite Algebras

Abstract

We study -representation finite -algebras obtained from tensor products of tensor algebras of species. In earlier work we computed the higher preprojective algebra of said algebras to be given as Jacobian algebras of certain species with potential , which are self-injective and finite dimensional. Truncating these Jacobian algebras yields a rich source of -representation finite -algebras. Under suitable assumptions, we prove that the set of all cuts of is transitive under successive cut-mutations. Furthermore, we show that cuts and cut-mutation correspond to truncated Jacobian algebras and -APR tilting, respectively. Consequently, under certain assumptions, all truncated Jacobian algebras are related to each other via -APR tilting. We produce various new examples of -representation finite -algebras.

Paper Structure

This paper contains 20 sections, 31 theorems, 96 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Outline
  3. Preliminaries
  4. Conventions
  5. Casimir Elements
  6. Species with Potential
  7. Jacobian Algebras
  8. d-Representation Finite Algebras
  9. Quiver with Cycles
  10. Quiver with Cycles and Cuts
  11. Canvas of a Quiver with Cycles
  12. Tensor Products and Quivers with Cycles
  13. Quiver with Cycles for Tensor Product
  14. Tensor Products of Species
  15. Truncated Jacobian Algebras
  16. ...and 5 more sections

Key Result

Theorem 1

(Theorem theorem - jacobian 2apr tils of each other) Let $(S, W)$ be a self-injective simply connected species with potential with enough cuts. Assume that $(S, W)$ has a preprojective cut. Then all cuts $C$ are preprojective and the corresponding truncated Jacobian algebras $\mathcal{P}(S, W, C)$ a

Figures (8)

  • Figure 1: Description of species over Dynkin diagrams
  • Figure 2: Dynkin Diagrams
  • Figure 3: Mutation lattice for $B_2\times B_2$
  • Figure 4: Mutation lattice for $A_3\times B_2$
  • Figure 5: Species of type $E_6\times F_4$
  • ...and 3 more figures

Theorems & Definitions (90)

  • Theorem
  • Lemma 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • Definition 2.6
  • Definition 2.7
  • Lemma 2.8
  • Definition 2.9
  • ...and 80 more