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Artin Algebras with Loops but No Outer Derivations

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Abstract

We construct families of Artin algebras over fields of arbitrary characteristic that contain a loop in their ordinary quiver but admit no nontrivial outer derivation. This refuses the long-held belief that such algebras should not exist.

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References

  1. Auslander, M., Reiten, I. and Smalø, S.: Representation Theory of Artin Algebras, Cambridge Stud. Adv. Math. 36, Cambridge Univ. Press, 1995.

  2. Bardzell, M. J. and Marcos, E. N.: Induced boundary maps for the cohomology of monomial and Auslander algebras, Canad. Math. Soc. Conf. Proc. 24 (1998), 47–54.

    Google Scholar 

  3. Bongartz, K. and Gabriel, P.: Covering spaces in representation theory, Invent. Math. 65 (1982), 331–378.

    Google Scholar 

  4. Cibils, C.: Review of D. Happel: 'Hochschild cohomology of finite-dimensional algebras', Séminaire d'algèbre P. Dubreil et M.-P. Malliavin, Proc., Paris/Fr. 1987/88, Lecture Notes in Math. 1404, Springer, New York, 1989, pp. 108–126 (MR: 91b:16012 (1991)).

    Google Scholar 

  5. Dirac, P. A. M.: The fundamental equations of quantum mechanics, Proc. Roy. Soc. A 109, p. 642; reprinted in: B. L. van der Warden (ed.), Sources of Quantum Mechanics, Dover Publ., New York, 1968.

  6. Gabriel, P.: Auslander-Reiten Sequences and Representation-Finite Algebras, Lecture Notes in Math. 831, Springer, New York, 1980, pp. 1–71.

    Google Scholar 

  7. Gastamina, S., de la Peña, J. A., Platzeck, M. I., Redondo, M. J. and Trepode, S.: Finite dimensional algebras with vanishing Hochschild cohomology, Preprint.

  8. Happel, D.: Hochschild cohomology of finite-dimensional algebras, Séminaire d'algèbre P. Dubreil et M.-P. Malliavin, Proc., Paris/Fr. 1987/88, Lecture Notes in Math. 1404, Springer, New York, 1989, pp. 108–126.

    Google Scholar 

  9. Igusa, K.: Notes on the no loop conjecture, J. Pure Appl. Algebra 69 (1990), 161–176.

    Google Scholar 

  10. Lenzing, H.: Nilpotente Elemente in Ringen von endlicher globaler Dimension, Math. Z. 108 (1969), 313–324.

    Google Scholar 

  11. Loday, J. L.: Cyclic Homology, Grundlehren Math. Wiss. 301, Springer, New York, 1992.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Toronto, Toronto, Ontario, Canada, M5S 3G3

    Ragnar-Olaf Buchweitz

  2. Département de mathématiques et d'informatique, Université de Sherbrooke, Sherbrooke, Québec, Canada, J1K 2R1

    Shiping Liu

Authors
  1. Ragnar-Olaf Buchweitz
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  2. Shiping Liu
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Buchweitz, RO., Liu, S. Artin Algebras with Loops but No Outer Derivations. Algebras and Representation Theory 5, 149–162 (2002). https://doi.org/10.1023/A:1015648905949

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