Cartan matrices of selfinjective algebras of tubular type
Jerzy Białkowski
Open Mathematics, Tome 2 (2004), p. 123-142 / Harvested from The Polish Digital Mathematics Library
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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 B^ /G of the repetitive algebras B^ of tubular algebras B with respect to the actions of admissible groups G of automorphisms of 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.

Publié le : 2004-01-01
EUDML-ID : urn:eudml:doc:268903 
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     year = {2004},
     pages = {123-142},
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Jerzy Białkowski. Cartan matrices of selfinjective algebras of tubular type. Open Mathematics, Tome 2 (2004) pp. 123-142. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_BF02475956/

[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. | Zbl 1092.16001

[2] D. Benson: “Representations and Cohomology I”, In: Cambridge Studies in Advanced Mathematics, Vol. 30, Cambridge, 1991.

[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 | Zbl 1023.16015

[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 | Zbl 1063.16020

[5] J. Białkowski and A. Skowroński: “Socle deformations of selfinjective algebras of tubular type”, J. Math. Soc. Japan, in press. | Zbl 1137.16024

[6] Yu.A. Drozd: “Tame and wild matrix problems”, In: Representation Theory II, Lecture Notes in Math., Vol. 832, Springer, Berlin-Heidelberg-New York, 1980; pp. 242–258. http://dx.doi.org/10.1007/BFb0088467

[7] K. Erdmann: “Algebras and quaternion defect groups I”, Math. Ann., Vol. 281, (1988), pp. 545–560. http://dx.doi.org/10.1007/BF01456838 | Zbl 0655.16011

[8] K. Erdmann: “Algebras and quaternion defect groups II”, Math. Ann., Vol. 281, (1988), pp. 561–582. http://dx.doi.org/10.1007/BF01456839 | Zbl 0655.16011

[9] K. Erdmann: Blocks of tame representation type and related algebras, Lecture Notes in Math., Vol. 1428, Springer, Berlin-Heidelberg-New York, 1990. | Zbl 0696.20001

[10] K. Erdmann, O. Kerner and A. Skowroński: “Self-injective algebras of wild tilted type”, J. Pure Appl. Algebra, Vol. 149, (2000), pp. 127–176. http://dx.doi.org/10.1016/S0022-4049(00)00035-9 | Zbl 0994.16015

[11] P. Gabriel: “The universal cover of a representation-finite algebra”, In: Representations of Algebras, Lecture Notes in Math., Vol. 903, Springer, Berlin-Heidelberg-New York, 1981, pp. 68–105. http://dx.doi.org/10.1007/BFb0092986

[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 | Zbl 0446.16032

[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. | Zbl 0488.16021

[15] H. Lenzing and A. Skowroński: “Roots of Nakayama and Auslander-Reiten translations”, Colloq. Math., Vol. 86, (2000), pp. 209–230. | Zbl 0982.16012

[16] J. Nehring and A. Skowroński: “Polynomial growth trivial extensions of simply connected algebras”, Fund. Math., Vol. 132, (1989), pp. 117–134. | Zbl 0677.16008

[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 | Zbl 0653.16021