Cartan matrices of selfinjective algebras of tubular type

Jerzy Białkowski

Open Mathematics (2004)

  • Volume: 2, Issue: 1, page 123-142
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Jerzy Białkowski. "Cartan matrices of selfinjective algebras of tubular type." Open Mathematics 2.1 (2004): 123-142. <http://eudml.org/doc/268903>.

@article{JerzyBiałkowski2004,
abstract = {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.},
author = {Jerzy Białkowski},
journal = {Open Mathematics},
keywords = {Primary: 15A15; 16D50; 16G60; Secondary 16G20; 16G70},
language = {eng},
number = {1},
pages = {123-142},
title = {Cartan matrices of selfinjective algebras of tubular type},
url = {http://eudml.org/doc/268903},
volume = {2},
year = {2004},
}

TY - JOUR
AU - Jerzy Białkowski
TI - Cartan matrices of selfinjective algebras of tubular type
JO - Open Mathematics
PY - 2004
VL - 2
IS - 1
SP - 123
EP - 142
AB - 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.
LA - eng
KW - Primary: 15A15; 16D50; 16G60; Secondary 16G20; 16G70
UR - http://eudml.org/doc/268903
ER -

References

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