Die Arbeit liefert eine abstrakt vollständige Klassifizierung endotrivialer Module über Präprojektiven Algebren von Dynkintypen ADE. Weiterhin gibt sie einen Ansatz für eine Verallgemeinerung. Weiterhin liefert sie einen Algorithmus um endotriviale Module zu konstruieren, deren Dimensionsvektoren positive Wurzeln sind. Für mehr Details verweisen wir auf die Einleitung der Arbeit.
Contents
Introduction
1 Modules over preprojective algebras
1.1 Prerequisites and basic definitions
1.2 Reflection functors
1.3 Endotrivial modules over preprojective algebras of Dynkin type ADE
1.4 Classification of endotrivial modules over the preprojective algebra of type Dn
2 Quivers with relations for symmetrizable Cartan matrices
2.1 Symmetrizable Cartan matrices and associated algebras
2.2 The Weyl group
2.3 Locally free H-modules
2.4 An analogy to the representation theory of modulated graphs
2.5 Representation theory of modulated graphs
2.6 Reflection functors
Objectives & Research Topics
The primary objective of this thesis is the classification of endotrivial modules over preprojective algebras associated with Dynkin quivers of type ADE, followed by an investigation into the generalization of these concepts for symmetrizable Cartan matrices of types Bn, Cn, F4, and G2 using representation-theoretic tools like reflection functors.
- Classification of endotrivial modules for ADE-type quivers.
- Application of reflection functors to construct modules.
- Extension of preprojective algebra definitions to symmetrizable Cartan matrices.
- Analysis of locally free modules and their representation theory.
- Inductive classification methods for endotrivial representations.
Excerpt from the Thesis
1.3 Endotrivial modules over preprojective algebras of Dynkin type ADE
In this section, we will see a very nice application of the reflection functors for the preprojective algebra, which strongly reminds us of the use of the Gelfand-Ponomarev reflection functors for the classification of indecomposables of path algebras of Dynkin type. Throughout this section, let Q be of Dynkin type ADE.
We start with Definition 1.12. We call a Π(Q)- module M endotrivial if it admits as endomorphisms only skalar multiples of the identity, i.e. EndΠ(Q)(M) ≅ k. We call S(Q) the set of endotrivial modules in mod- Π(Q).
We directly see that every endotrivial module M is indecomposable, since EndΠ(Q)(M) ≅ k is in particular a local ring. Vice versa, not every indecomposable module must be endotrivial. An example is given by the Π(A3)- module M represented by
2
1 3
2 .
Recall, that in order to satisfy the preprojective relations, we have to put a minus sign somewhere if we want to express this in terms of matrices. An easy calculation shows, that EndΠ(A3)(M) is isomorphic to k[X]/(X2), hence M is not endotrivial.
Furthermore, Lemma 1.2 forces an endotrivial module M to be rigid, i.e. we have Ext1 Π(Q)(M,M) = 0. This follows by a simple calculation. We have dim Ext1 Π(Q)(M,M) = dim HomΠ(Q)(M,M) + dim HomΠ(Q)(M,M) − (dim(M), dim(M)) = 1 + 1 − 2qQ(dim(M)), where qQ is the Tits form of Q, which is positive definite, since we are in the Dynkin case. Thus, we get 0 ≤ dim Ext1 Π(Q)(M,M) ≤ 0. Note, that the example above, where M is the projective module P2, shows that not every rigid module has to be endotrivial. Furthermore, Lemma 1.2 implies with the same calculation that, for an endotrivial module M, we have qQ(dimM) = 1, which means that dim(M) is a root, hence it is the dimension vector of an indecomposable kQ-module. For later reference, we state this as a Lemma 1.13. Let M ∈ mod- Π(Q) be endotrivial, Q be of Dynkin type ADE. Then dim(M) is a root of the Tits form qQ and M is rigid.
Summary of Chapters
Introduction: Outlines the goal of classifying endotrivial modules over preprojective algebras for Dynkin quivers of type ADE and introduces key formulas.
1 Modules over preprojective algebras: Covers prerequisites, introduces reflection functors, and provides a classification for ADE and Dn types.
1.1 Prerequisites and basic definitions: Establishes fundamental definitions of quivers, path algebras, and preprojective relations.
1.2 Reflection functors: Defines reflection functors on modules and studies their properties, specifically for the Dynkin case.
1.3 Endotrivial modules over preprojective algebras of Dynkin type ADE: Classifies endotrivial modules and proves properties regarding their rigidity and dimension vectors.
1.4 Classification of endotrivial modules over the preprojective algebra of type Dn: Applies theoretical results to classify endotrivial modules specifically for type Dn.
2 Quivers with relations for symmetrizable Cartan matrices: Extends the preprojective algebra definition to more general Cartan matrices.
2.1 Symmetrizable Cartan matrices and associated algebras: Introduces definitions for symmetrizable generalized Cartan matrices and the algebra H.
2.2 The Weyl group: Discusses the Weyl group W(C) and the set of real roots ΔRe(C) associated with Cartan matrices.
2.3 Locally free H-modules: Defines locally free modules and establishes the homological bilinear forms for H.
2.4 An analogy to the representation theory of modulated graphs: Treats representations of H as a form of k-species to simplify module calculations.
2.5 Representation theory of modulated graphs: Proves that the category of representations is abelian and isomorphic to the category of representations of H.
2.6 Reflection functors: Extends the definition of reflection functors to general preprojective algebras Π(C, D).
Keywords
Preprojective algebra, Endotrivial modules, Reflection functors, Dynkin quiver, Symmetrizable Cartan matrices, Representation theory, Rigid modules, Modulated graphs, Weyl group, Auslander-Reiten theory, Dimension vector, Tits form, Locally free modules.
Frequently Asked Questions
What is the primary focus of this thesis?
The thesis focuses on the classification of endotrivial modules over preprojective algebras, first for Dynkin quivers of type ADE, and then extending the study to symmetrizable Cartan matrices.
What are endotrivial modules?
Endotrivial modules are modules over a preprojective algebra whose endomorphism ring consists only of scalar multiples of the identity.
Which mathematical tools are central to the classification?
Reflection functors are the primary tool used to construct modules and prove classification theorems in both the ADE and the general Cartan matrix cases.
How is the classification for ADE types structured?
It involves characterizing dimension vectors as roots of the Tits form and using inductive construction via reflection functors starting from simple modules.
What does the second part of the thesis introduce?
The second part introduces a generalization of preprojective algebras defined via symmetrizable generalized Cartan matrices to cover Dynkin types like Bn, Cn, F4, and G2.
What is the role of the Weyl group in this research?
The Weyl group is used to describe the root systems associated with the Cartan matrices and helps in understanding the reflection properties of the representations.
How does the thesis handle the representation theory of modulated graphs?
It establishes an isomorphism between the representation category of the modulated graph and the category of modules over the generalized preprojective algebra, facilitating the transfer of results.
Is the classification for type Dn complete?
Yes, the thesis provides an explicit classification of endotrivial modules for type Dn using an inductive approach based on the results for D4.
- Citar trabajo
- Jakob Bongartz (Autor), 2015, Endotrivial modules over preprojective algebras, Múnich, GRIN Verlag, https://www.grin.com/document/335106
