Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The preprojective algebra of a finite EI quiver

Dongdong Hu

Abstract

We define the preprojective algebra of a finite EI quiver. We prove that it is isomorphic to a centain tensor algebra. For a finite EI quiver of Cartan type, we prove that the corresponding preprojective algebra is isomorphic to the generalized preprojective algebra.

The preprojective algebra of a finite EI quiver

Abstract

We define the preprojective algebra of a finite EI quiver. We prove that it is isomorphic to a centain tensor algebra. For a finite EI quiver of Cartan type, we prove that the corresponding preprojective algebra is isomorphic to the generalized preprojective algebra.

Paper Structure

This paper contains 12 sections, 15 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Finite EI quivers and preprojective algebras
  3. The preprojective algebra
  4. The algebra $\Pi$ as a tensor algebra
  5. Representations of a finite EI quiver
  6. Locally projective representations
  7. Trace maps
  8. Bimodules and the preprojective algebra
  9. The bimodule $\Pi(Q,X)_1$
  10. Auslander-Reiten translations
  11. The main result
  12. The algebras associated to Cartan matrices

Key Result

Proposition 2.1

$($CW$)$ Let $\mathcal{C}=\mathcal{C}(Q,X)$ be the category associated to a finite EI quiver $(Q,X)$. Set $A=\prod_{x\in Q_0}\mathbb{K}X(x)$ and $V=\bigoplus_{\alpha\in Q_1}\mathbb{K}X(\alpha)$, where $V$ is naturally an $A$-$A$-bimodule. Then there is an algebra isomorphism $T_A(V)\rightarrow\mathb

Theorems & Definitions (32)

  • Proposition 2.1
  • Definition 2.2
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • proof
  • Proposition 3.1
  • proof
  • ...and 22 more