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The Grothendieck Groups and Stable Equivalences of Mesh Algebras

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Abstract

We deal with the finite-dimensional mesh algebras given by stable translation quivers. These algebras are self-injective, and thus the stable module categories have a structure of triangulated categories. Our main result determines the Grothendieck groups of these stable module categories. As an application, we give a complete classification of the mesh algebras up to stable equivalences.

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References

  1. Adachi, T., Iyama, O., Reiten, I.: t-tilting theory. Compos. Math. 150, 415–452 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Angeleri Hügel, D., Happel, D., Krause, H.: Handbook of tilting theory, London Mathematical Society Lecture Note Series, 332. Cambridge university press (2007)

  3. Andreu Juan, E., Saorín, M.: The symmetry, period and Calabi–Yau dimension of finite dimensional mesh algebras. Journal of Algebra 429, 19–74 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Assem, I., Simson, D., Skowronski, A.: Elements of the Representation Theory of Associative Algebras: Volume 1: Techniques of Representation Theory, London Mathematical Society Student Texts, 65. Cambridge University Press (2006)

  5. Auslander, M., Reiten, I.: D Tr -periodic modules and functors. In: CMS Conf. Proc. ICRA VII (Cocoyoc 1994), vol. 18, pp. 39–50 (1996)

  6. Björner, A., Brenti, F.: Combinatorics of Coxeter Groups, Graduate Texts in Mathematics, 231. Springer, New York (2005)

    MATH  Google Scholar 

  7. Buan, A.B., Iyama, O., Reiten, I., Scott, J.: Cluster structures for 2-Calabi–Yau categories and unipotent groups. Compos. Math. 145, 1035–1079 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dugas, A.: Resolutions of mesh algebras: periodicity and Calabi–Yau dimensions. Math. Z. 271, 1151–1184 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gabriel, P.: The universal cover of a representation-finite algebra, Representations of Algebras, Lecture Notes in Math., vol. 903, pp. 68–105. Springer, Berlin (1981)

    Google Scholar 

  10. Happel, D.: On the derived category of a finite-dimensional algebra, Comment. Math. Helv. 62, 339–389 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  11. Happel, D.: Triangulated Categories in the Representation Theory of Finite Dimensional Algebras, London Mathematical Society Lecture Note Series, 119. Cambridge University Press (1988)

  12. Holm, T., Jørgensen, P., Rouquier, R.: Triangulated Categories, London Mathematical Society Lecture Note Series, 375. Cambridge University Press (2010)

  13. Iyama, O.: Auslander correspondence. Adv. Math. 210, 51–82 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Iyama, O., Yoshino, Y.: Mutation in triangulated categories and rigid Cohen-Macaulay modules. Invent. Math. 172(1), 117–168 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Keller, B., Reiten, I.: Cluster-tilted algebras are Gorenstein and stably Calabi–yau. Adv. Math. 211(1), 123–151 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. König, S., Zimmermann, A.: Derived Equivalences for Group Rings, Lecture Notes in Mathematics, vol. 1685. Springer, Berlin (1998)

    Google Scholar 

  17. Rickard, J.: Derived categories and stable equivalence. Journal of Pure and Applied Algebra 61(3), 303–317 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  18. Rickard, J.: Morita theory for derived categories. J. London Math. Soc. s2-39 (3), 436–456 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  19. Riedtmann, C.: Algebren, Darstellungsköcher, Überlagerungen und zurück. (German), Comment. Math. Helv. 55(2), 199–224 (1980)

    Article  MATH  Google Scholar 

  20. Zhou, Y., Zhu, B.: Maximal rigid subcategories in 2-Calabi–Yau triangulated categories. Journal of Algebra 348(1), 49–60 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The author is a Research Fellow of Japan Society for the Promotion of Science (JSPS). This work was supported by JSPS KAKENHI Grant Number JP16J02249.

This paper is based on the master thesis of the author in Nagoya University.

The author thanks his supervisor Osamu Iyama for thorough instruction, and Manuel Saorín for answering my questions and giving me useful advice. He also thanks the anonymous reviewer for helping me to correct mistakes and make this paper more readable.

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Authors and Affiliations

  1. Graduate School of Mathematics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya-shi, Aichi-ken, 464-8602, Japan

    Sota Asai

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  1. Sota Asai
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Correspondence to Sota Asai.

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Presented by Vlastimil Dlab.

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Asai, S. The Grothendieck Groups and Stable Equivalences of Mesh Algebras. Algebr Represent Theor 21, 635–681 (2018). https://doi.org/10.1007/s10468-017-9732-x

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  • DOI: https://doi.org/10.1007/s10468-017-9732-x

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