Subtopic Deep Dive
Quiver Representations
Research Guide
What is Quiver Representations?
Quiver representations study modules over path algebras of directed graphs called quivers, focusing on classification of indecomposables via Auslander-Reiten theory and connections to cluster categories.
Quiver representations classify finite-dimensional algebras through combinatorial quiver data and path algebras (Gabriel, 1972). Key developments include triangulated orbit categories and cluster-tilting objects in 2-Calabi-Yau categories. Over 1,500 papers cite foundational works like Keller (2005, 350 citations) and Derksen et al. (2010, 332 citations).
Why It Matters
Quiver representations classify indecomposable modules for hereditary algebras, enabling explicit computations in representation theory (Keller, 2005). They underpin cluster algebras, linking combinatorics to categorification and total positivity (Derksen, Weyman, Zelevinsky, 2010; Buan et al., 2009). Applications appear in algebraic geometry via derived categories and in physics through Calabi-Yau categories modeling string theory compactifications (Amiot, 2009).
Key Research Challenges
Classifying indecomposables
Determining all indecomposable representations for wild quivers remains open due to infinite families. Auslander-Reiten theory provides quivers of type A-D-E but fails for wild cases (Keller, 2005). Tame-wild dichotomy classification relies on combinatorial invariants.
Cluster tilting extensions
Extending cluster-tilting objects beyond Dynkin quivers to non-Dynkin cases challenges Calabi-Yau categories. Buan et al. (2009) construct structures for preprojective algebras, but mutations require new silting objects (Koenig, Yang, 2014).
Potential stability
Quivers with potentials demand Jacobian stability for representation equivalence. Derksen et al. (2010) apply this to cluster algebras, but verifying stability computationally scales poorly for large quivers.
Essential Papers
On triangulated orbit categories
Bernhard Keller · 2005 · Documenta Mathematica · 350 citations
We show that the category of orbits of the bounded derived category of a hereditary category under a well-behaved autoequivalence is canonically triangulated. This answers a question by Aslak Buan,...
Quivers with potentials and their representations II: Applications to cluster algebras
Harm Derksen, Jerzy Weyman, Andrei Zelevinsky · 2010 · Journal of the American Mathematical Society · 332 citations
We continue the study of quivers with potentials and their representations initiated in the first paper of the series. Here we develop some applications of this theory to cluster algebras. As shown...
On differential graded categories
Bernhard Keller · 2007 · Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006 · 307 citations
Differential graded categories enhance our understanding of triangulated categories appearing in algebra and geometry. In this survey, we review their foundations and report on recent work by Drinf...
Cluster structures for 2-Calabi–Yau categories and unipotent groups
Aslak Bakke Buan, Osamu Iyama, Idun Reiten et al. · 2009 · Compositio Mathematica · 273 citations
Abstract We investigate cluster-tilting objects (and subcategories) in triangulated 2-Calabi–Yau and related categories. In particular, we construct a new class of such categories related to prepro...
From triangulated categories to cluster algebras II
Ph. Caldero, Barbara Keller · 2006 · Annales Scientifiques de l École Normale Supérieure · 247 citations
In the acyclic case, we establish a one-to-one correspondence between the tilting objects of the cluster category and the clusters of the associated cluster algebra. This correspondence enables us ...
Cluster mutation via quiver representations
Aslak Bakke Buan, Bethany Marsh, Idun Reiten · 2008 · Commentarii Mathematici Helvetici · 160 citations
Matrix mutation appears in the definition of cluster algebras of Fomin and Zelevinsky. We give a representation theoretic interpretation of matrix mutation, using tilting theory in cluster categori...
Silting objects, simple-minded collections, $t$-structures and co-$t$-structures for finite-dimensional algebras
Steffen Koenig, Dong Yang · 2014 · Documenta Mathematica · 157 citations
Bijective correspondences are established between (1) silting objects, (2) simple-minded collections, (3) bounded t -structures with length heart and (4) bounded co- t -structures. These correspond...
Reading Guide
Foundational Papers
Start with Keller (2005) for triangulated orbit categories establishing cluster category foundations; then Derksen et al. (2010) for quivers with potentials linking to cluster algebras.
Recent Advances
Koenig, Yang (2014) on silting objects unifying t-structures; Caldero, Keller (2006) connecting tilting to clusters.
Core Methods
Path algebras, AR-duality, tilting modules, cluster tilting in 2-CY categories, Jacobian algebras from potentials.
How PapersFlow Helps You Research Quiver Representations
Discover & Search
Research Agent uses citationGraph on Keller (2005) to map 350+ citations linking quiver reps to cluster categories, then findSimilarPapers reveals Buan et al. (2009) for 2-Calabi-Yau extensions. exaSearch queries 'quiver representations tilting theory' yielding 200+ papers with AR-quiver visuals.
Analyze & Verify
Analysis Agent runs readPaperContent on Derksen et al. (2010) extracting quiver potential definitions, then verifyResponse (CoVe) with GRADE grading confirms cluster mutation claims against 50 citations. runPythonAnalysis computes representation dimensions via NumPy for indecomposable classification verification.
Synthesize & Write
Synthesis Agent detects gaps in wild quiver classifications across 20 papers, flagging open problems. Writing Agent applies latexEditText to AR-quiver diagrams, latexSyncCitations for 15 refs, and latexCompile for publication-ready notes; exportMermaid generates mutation graphs from cluster data.
Use Cases
"Compute dimension vectors for indecomposables of A_5 quiver."
Research Agent → searchPapers('A_n quiver representations') → Analysis Agent → runPythonAnalysis (NumPy quiver module simulator) → dimension table and AR-quiver plot.
"Draft section on cluster mutations with citations."
Synthesis Agent → gap detection (Buan, Marsh, Reiten 2008) → Writing Agent → latexEditText + latexSyncCitations (10 refs) + latexCompile → LaTeX PDF with mutation diagrams.
"Find code for quiver potential stability."
Research Agent → paperExtractUrls (Derksen et al. 2010) → Code Discovery → paperFindGithubRepo → githubRepoInspect → SageMath quiver rep code with Jacobian algebra examples.
Automated Workflows
Deep Research scans 50+ papers from Keller (2005) citationGraph, producing structured report on tilting theory evolution with GRADE-verified claims. DeepScan applies 7-step CoVe to Amiot (2009), verifying 2-CY category constructions via runPythonAnalysis. Theorizer generates hypotheses on silting extensions from Koenig-Yang (2014) patterns.
Frequently Asked Questions
What defines quiver representations?
Modules over path algebras of finite directed graphs (quivers), classified by dimension vectors and Auslander-Reiten quivers for tame cases.
What methods classify representations?
Gabriel quivers for finite representation type; tilting theory and cluster categories for categorification (Keller 2005; Buan et al. 2009).
What are key papers?
Keller (2005, 350 cites) on triangulated orbit categories; Derksen, Weyman, Zelevinsky (2010, 332 cites) on quivers with potentials for clusters.
What open problems exist?
Full classification of wild quiver representations; extending cluster structures beyond Dynkin types (Ingalls, Thomas 2009).
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