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2004 | 2 | 1 | 123-142
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Cartan matrices of selfinjective algebras of tubular type

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The Cartan matrix of a finite dimensional algebra A is an important combinatorial invariant reflecting frequently structural properties of the algebra and its module category. For example, one of the important features of the modular representation theory of finite groups is the nonsingularity of Cartan matrices of the associated group algebras (Brauer’s theorem). Recently, the class of all tame selfinjective algebras having simply connected Galois coverings and the stable Auslander-Reiten quiver consisting only of stable tubes has been shown to be the class of selfinjective algebras of tubular type, that is, the orbit algebras $$ \hat B$$ /G of the repetitive algebras $$ \hat B$$ of tubular algebras B with respect to the actions of admissible groups G of automorphisms of $$ \hat B$$ . The aim of the paper is to describe the determinants of the Cartan matrices of selfinjective algebras of tubular type and derive some consequences.
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Czasopismo
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Tom
2
Numer
1
Strony
123-142
Opis fizyczny
Daty
wydano
2004-03-01
online
2004-03-01
Twórcy
Bibliografia
  • [1] I. Assem, D. Simson, and A. Skowroński: “Elements of Representation Theory of Associative Algebras. I: Techniques of Representation Theory”, In: London Mathematical Society Student Texts, Cambridge University Press, in press.
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  • [3] J. Białkowski and A. Skowroński: “Selfinjective algebras of tubular type”, Colloq. Math., Vol. 94, (2002), pp. 175–194. http://dx.doi.org/10.4064/cm94-2-2
  • [4] J. Białkowski and A. Skowroński: “On tame weakly symmetric algebras having only periodic modules”, Archiv. Math., Vol. 81, (2003), pp. 142–154. http://dx.doi.org/10.1007/s00013-003-0816-y
  • [5] J. Białkowski and A. Skowroński: “Socle deformations of selfinjective algebras of tubular type”, J. Math. Soc. Japan, in press.
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  • [12] D. Happel, U. Preiser and C.M. Ringel: “Vinberg’s characterization of Dynkin diagrams using subadditive functions with application to D Tr-periodic modules”, In: Representation Theory II., Lecture Notes in Math., Vol. 832, Springer, Berlin-Heidelberg-New York, 1980, pp. 280–294. http://dx.doi.org/10.1007/BFb0088469
  • [13] D. Happel and C.M. Ringel: “The derived category of a tubular algebra”, In: Representation Theory I, Lecture Notes in Math., Vol. 1177, Springer, Berlin-Heidelberg-New York, 1986, pp. 156–180.
  • [14] D. Hughes and J. Waschbüsch: “Trivial extensions of tilted algebras”, Proc. London Math. Soc., Vol. 46, (1983), pp. 347–364.
  • [15] H. Lenzing and A. Skowroński: “Roots of Nakayama and Auslander-Reiten translations”, Colloq. Math., Vol. 86, (2000), pp. 209–230.
  • [16] J. Nehring and A. Skowroński: “Polynomial growth trivial extensions of simply connected algebras”, Fund. Math., Vol. 132, (1989), pp. 117–134.
  • [17] C.M. Ringel: Tame Algebras and Integral Quadratic Forms, Lecture Notes in Math., Vol. 1099, Springer, Berlin-Heidelberg-New York, 1984.
  • [18] A. Skowroński: “Selfinjective algebras of polynomial growth”, Math. Annalen, Vol. 285, (1989), pp. 177–199. http://dx.doi.org/10.1007/BF01443513
Typ dokumentu
Bibliografia
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bwmeta1.element.doi-10_2478_BF02475956
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