Inventiones mathematicae

, Volume 209, Issue 1, pp 61–158 | Cite as

Quivers with relations for symmetrizable Cartan matrices I: Foundations

  • Christof Geiss
  • Bernard LeclercEmail author
  • Jan Schröer


We introduce and study a class of Iwanaga–Gorenstein algebras defined via quivers with relations associated with symmetrizable Cartan matrices. These algebras generalize the path algebras of quivers associated with symmetric Cartan matrices. We also define a corresponding class of generalized preprojective algebras. For these two classes of algebras we obtain generalizations of classical results of Gabriel, Dlab–Ringel, and Gelfand–Ponomarev. In particular, we obtain new representation theoretic realizations of all finite root systems without any assumption on the ground field.

Mathematics Subject Classification

Primary 16G10 16G20 Secondary 16G70 



We thank the CIRM (Luminy) for two weeks of hospitality in July 2013, where this work was initiated. The first author acknowledges financial support from UNAM-PAPIIT Grant IN108114. The third author thanks the SFB/Transregio TR 45 for financial support, and the UNAM for one month of hospitality in March 2014. We thank W. Crawley-Boevey, H. Lenzing and C.M. Ringel for helpful comments.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Christof Geiss
    • 1
  • Bernard Leclerc
    • 2
    • 3
    Email author
  • Jan Schröer
    • 4
  1. 1.Instituto de MatemáticasUniversidad Nacional Autónoma de México Ciudad UniversitariaMexicoMexico
  2. 2.LMNONormandie Univ, UNICAEN, CNRSCaenFrance
  3. 3.Institut Universitaire de FranceParis Cedex 05France
  4. 4.Mathematisches InstitutUniversität BonnBonnGermany

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