Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Jordan recoverability of some subcategories of modules over gentle algebras

Benjamin Dequêne

Abstract

Gentle algebras form a class of finite-dimensional algebras introduced by I. Assem and A. Skowroński in the 1980s. Modules over such an algebra can be described by string and band combinatorics in the associated gentle quiver from the work of M.C.R. Butler and C.M. Ringel. Any module can be naturally associated to a quiver representation. A nilpotent endomorphism of a quiver representation induces linear transformations over vector spaces at each vertex. Generically among all nilpotent endomorphisms, a well-defined Jordan form exists for these representations. We focus on subcategories additively generated by all the indecomposable representations of a gentle quiver, including a fixed vertex in their support. We show a characterization of the vertices such that the objects of this subcategory are determined up to isomorphism by their generic Jordan form.

Jordan recoverability of some subcategories of modules over gentle algebras

Abstract

Gentle algebras form a class of finite-dimensional algebras introduced by I. Assem and A. Skowroński in the 1980s. Modules over such an algebra can be described by string and band combinatorics in the associated gentle quiver from the work of M.C.R. Butler and C.M. Ringel. Any module can be naturally associated to a quiver representation. A nilpotent endomorphism of a quiver representation induces linear transformations over vector spaces at each vertex. Generically among all nilpotent endomorphisms, a well-defined Jordan form exists for these representations. We focus on subcategories additively generated by all the indecomposable representations of a gentle quiver, including a fixed vertex in their support. We show a characterization of the vertices such that the objects of this subcategory are determined up to isomorphism by their generic Jordan form.
Paper Structure (8 sections, 36 theorems, 3 equations)

This paper contains 8 sections, 36 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Gentle algebras and representations
  4. Jordan recoverability and canonical Jordan recoverability
  5. Proof of the main results
  6. Proof of Theorem \ref{['main']}
  7. Proof of Theorem \ref{['2ndmain']}
  8. To go further

Key Result

Theorem 1.4

If $Q$ is a Dynkin quiver, and $m$ is a minuscule vertex, then $\mathscr{C}_{\mathcal{Q},m}$ is canonically Jordan recoverable.

Theorems & Definitions (97)

  • Example 1.1
  • Example 1.2
  • Example 1.3
  • Theorem 1.4: GPT19
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 2.1
  • Example 2.2
  • Remark 2.3
  • ...and 87 more