Hostname: page-component-76fb5796d-2lccl Total loading time: 0 Render date: 2024-04-25T22:42:04.528Z Has data issue: false hasContentIssue false
  • English
  • Français

GROUP ACTIONS AND COVERINGS OF BRAUER GRAPH ALGEBRAS

Published online by Cambridge University Press:  30 August 2013

EDWARD L. GREEN ,
SIBYLLE SCHROLL  and
NICOLE SNASHALL
EDWARD L. GREEN
Affiliation:
Department of Mathematics, Virginia Tech, Blacksburg, VA 24061-0123, USA e-mail: green@math.vt.edu
SIBYLLE SCHROLL
Affiliation:
Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, United Kingdom e-mail: ss489@le.ac.uk
NICOLE SNASHALL
Affiliation:
Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, United Kingdom e-mail: njs5@mcs.le.ac.uk
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We develop a theory of group actions and coverings on Brauer graphs that parallels the theory of group actions and coverings of algebras. In particular, we show that any Brauer graph can be covered by a tower of coverings of Brauer graphs such that the topmost covering has multiplicity function identically one, no loops, and no multiple edges. Furthermore, we classify the coverings of Brauer graph algebras that are again Brauer graph algebras.

Keywords

Primary 05E1816G20Secondary 14E2016W5058E40
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 56 , Issue 2 , May 2014 , pp. 439 - 464
Copyright
Copyright © Glasgow Mathematical Journal Trust 2013 

References

REFERENCES

1.Alperin, J. L., Local representation theory: Modular representations as an introduction to the local representation theory of finite groups, Cambridge Studies in Advanced Mathematics, vol. 11 (Cambridge University Press, Cambridge, UK, 1986).Google Scholar
2.Asashiba, H., A generalization of Gabriel's galois covering functors and derived equivalences, J. Algebra 334 (2011), 109–149.Google Scholar
3.Benson, D. J., Representations and cohomology. I. Basic representation theory of finite groups and associative algebras, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 30 (Cambridge University Press, Cambridge, UK, 1998).Google Scholar
4.Bongartz, K. and Gabriel, P., Covering spaces in representation-theory, Invent. Math. 65 (1981–82), 331–378.CrossRefGoogle Scholar
5.Brauer, R., Investigations on group characters, Ann. Math. 42 (1941), 936–958.Google Scholar
6.Cibils, C. and Marcos, E. N., Skew category, Galois covering and smash product of a (k)-category, Proc. Amer. Math. Soc. 134 (2006), 39–50.Google Scholar
7.Gordon, R. and Green, E. L., Graded Artin algebras, J. Algebra 76 (1982), 111–137.Google Scholar
8.Green, E. L., Graphs with relations, coverings and group-graded algebras, Trans. Amer. Math. Soc. 279 (1983), 297–310.Google Scholar
9.Green, E. L., Hunton, J. R. and Snashall, N., Coverings, the graded center and Hochschild cohomology, J. Pure Appl. Algebra 212 (2008), 2691–2706.CrossRefGoogle Scholar
10.Green, E. L., Schroll, S., Snashall, N. and Taillefer, R., The Ext algebra of a Brauer graph algebra, preprint (arXiv:1302.6413).Google Scholar
11.Green, J. A., Walking around the Brauer tree, J. Austral. Math. Soc. 17 (1974), 197–213 (collection of articles dedicated to the memory of Hanna Neumann, VI).Google Scholar
12.Janusz, G., Indecomposable modules for finite groups, Ann. Math. 89 (1969), 209–241.Google Scholar
13.Kauer, M., Derived equivalences of graph orders, PhD Thesis (Universität Stuttgart, 1996).Google Scholar
14.Kauer, M., Derived equivalence of graph algebras, in Trends in the representation theory of finite-dimensional algebras, Contemporary Mathematics, vol. 229 (L, Green E., Editor) (American Mathematical Society, Providence, RI, 1998), 201–213.Google Scholar
15.Kauer, M. and Roggenkamp, K. W., Higher-dimensional orders, graph-orders, and derived equivalences, J. Pure Appl. Algebra 155 (2001), 181–202.Google Scholar
16.Membrillo-Hernández, F. H., Brauer tree algebras and derived equivalence, J. Pure Appl. Algebra 114 (1997), 231–258.Google Scholar
17.Reiten, I., Almost split sequences for group algebras of finite representation type, Trans. Amer. Math. Soc. 233 (1977), 125–136.Google Scholar
18.Rickard, J., Derived categories and stable equivalence, J. Pure Appl. Algebra 61 (1989), 303–317.CrossRefGoogle Scholar
19.Riedtmann, Ch., Algebren, Darstellungsköcher, Überlagerungen und zurück, Comment. Math. Helv. 55 (1980), 199–224.CrossRefGoogle Scholar
20.Roggenkamp, K. W., Biserial algebras and graphs, in Algebras and modules, II, CMS Conf. Proc., vol. 24 (Reiten, I., Smalø, S. O. and Solberg, Ø., Editors) (American Mathematical Society, Providence, RI, 1998), 481–496.Google Scholar
21.Skowroński, A., Selfinjective algebras: Finite and tame type, in Trends in representation theory of algebras and related topics, Contemporary Mathematics, vol. 406 (de la Peña, J. A. and Bautista, R., Editors) (American Mathematical Society, Providence, RI, 2006), 169–238.Google Scholar
22.Skowroński, A., Yamagata, K., Positive Galois coverings of self-injective algebras, Adv. Math. 194 (2005), 398–436.Google Scholar