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Derived Picard groups of selfinjective Nakayama algebras

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Abstract

In our preceding paper a generating set of the derived Picard group of a selfinjective Nakayama algebra was constructed combining some previous results for Brauer tree algebras and the technique of orbit categories developed there. In this paper we finish the computation of the derived Picard group of a selfinjective Nakayama algebra.

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References

  1. Keller, B.: Hochschild cohomology and derived Picard groups. J. Pure Appl. Algebra 190, 177–196 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Huisgen-Zimmermann, B., Saorín, M.: Geometry of chain complexes and outer automorphisms under derived equivalence. Trans. Am. Math. Soc. 353(12), 4757–4777 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Rouquier, R.: Automorphismes, graduations et catégories triangulées. J. Inst. Math. Jussieu 10(3), 713–751 (2011)

    Article  MathSciNet  Google Scholar 

  4. Rouquier, R., Zimmermann, A.: Picard groups for derived module categories. Proc. Lond. Math. Soc. 87(1), 197–225 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Yekutieli, A.: Dualizing complexes, Morita equivalence and the derived Picard group of a ring. J. Lond. Math. Soc. 60(3), 723–746 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  6. Miyachi, J.I., Yekutieli, A.: Derived Picard groups of finite-dimensional hereditary algebras. Compost. Math. 129(3), 341–368 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Lenzing, H., Meltzer, H.: The automorphism group of the derived category for a weighted projective line. Commun. Algebra 28(4), 1685–1700 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Broomhead, N., Pauksztello, D., Ploog, D.: Discrete derived categories I: homomorphisms, autoequivalences and t-structures. (2013). arXiv:1312.5203

  9. Zimmermann, A.: Self-equivalences of the derived category of Brauer tree algebras with exceptional vertex. An. Ştiinţ. Univ. Ovidius Constanţa Ser. Mat. 9(1), 139–148 (2001)

    MathSciNet  MATH  Google Scholar 

  10. Schaps, M., Zakay-Illouz, E.: Braid group action on the refolded tilting complex of the Brauer star algebra. Proc. ICRA IX Beijing 2, 434–449 (2002)

    MATH  Google Scholar 

  11. Seidel, P., Thomas, R.: Braid group actions on derived categories of coherent sheaves. Duke Math. J. 108(1), 37–108 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. Khovanov, M., Seidel, P.: Quivers, Floer cohomology, and braid group actions. J. Am. Math. Soc. 15(1), 203–271 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Muchtadi-Alamsyah, I.: Braid action on derived category of Nakayama algebras. Commun. Algebra 36(7), 2544–2569 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Zvonareva, A.: Mutations and the derived Picard group of the Brauer star algebra. J. of Algebra. 443, 270–299 (2015)

  15. Volkov, Y., Zvonareva, A.: On standard derived equivalences of orbit categories. (2015). arXiv:1501.00709

  16. Efimov, A.: Braid group actions on triangulated categories. The Möbius contest paper (2007)

  17. Rickard, J.: Derived equivanences as derived functors. J. Lond. Math. Soc. 43(1), 37–48 (1991)

    Article  MATH  Google Scholar 

  18. Keller, B.: Deriving DG categories. Ann. Sci. École Norm. Sup. (4) 27(1), 63–102 (1994)

    MathSciNet  MATH  Google Scholar 

  19. Rickard, J.: Morita theory for derived categories. J. Lond. Math. Soc. 39(3), 436–456 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  20. Bolla, M.: Isomorphisms between endomorphism rings of progenerators. J. Algebra 87(1), 261–281 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  21. Linckelmann, M.: Stable equivalences of Morita type for self-injective algebras and \(p\)-groups. Math. Z. 223(1), 87–100 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  22. Pollack, R.D.: Algebras and their automorphism groups. Commun. Algebra 17(8), 1843–1866 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  23. Guil-Asensio, F., Saorín, M.: The group of outer automorphisms and the Picard group of an algebra. Algebra Represent. Theory 2(4), 313–330 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  24. Erdmann, K., Holm, T.: Twisted bimodules and Hochschild cohomology for self-injective algebras of class \(A_n\). Forum Math. 11(2), 177–201 (1999)

    MathSciNet  MATH  Google Scholar 

  25. Brieskorn, E., Saito, K.: Artin-gruppen und Coxeter-gruppen. Invent. Math. 17(4), 245–271 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  26. Charney, R., Peifer, D.: The \(K(\pi,1)\)-conjecture for the affine braid group. Comment. Math. Helv. 78(3), 584–600 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. Allcock, D.: Braid pictures for Artin groups. Trans. Am. Math. Soc. 354(9), 3455–3474 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  28. Birman, J.S.: Mapping class groups and their relationship to braid groups. Commun. Pure Appl. Math. 22(2), 213–238 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  29. Lyndon, R.C., Schupp, P. E.: Combinatorial group theory. Ergebnisse der Math., Springer, Berlin 89 (1977)

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Correspondence to Yury Volkov.

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To the memory of Alexander Ivanov

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Volkov, Y., Zvonareva, A. Derived Picard groups of selfinjective Nakayama algebras. manuscripta math. 152, 199–222 (2017). https://doi.org/10.1007/s00229-016-0859-6

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  • DOI: https://doi.org/10.1007/s00229-016-0859-6

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