Hostname: page-component-7b7567c8c-tnzcf Total loading time: 0 Render date: 2026-11-14T12:17:49.976Z Has data issue: false hasContentIssue false
  • English
  • Français

Strongly simply connected Auslander algebras

Published online by Cambridge University Press:  18 May 2009

Ibrahim Assem and
Peter Brown
Ibrahim Assem
Affiliation:
Département de Mathématiques et d'Informatique, Université de Sherbrooke, Sherbrooke (Québec) J1K 2R1, Canada
Peter Brown
Affiliation:
Département de Mathématiques et d'Informatique, Université de Sherbrooke, Sherbrooke (Québec) J1K 2R1, Canada
Rights & Permissions [Opens in a new window]

Extract

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.

Let k be an algebraically closed field. By an algebra is meant an associative finite dimensional k-algebra A with an identity. We are interested in studying the representation theory of Λ, that is, in describing the category mod Λ of finitely generated right Λ-modules. Thus we may, without loss of generality, assume that Λ is basic and connected. For our purpose, one strategy consists in using covering techniques to reduce the problem to the case where the algebra is simply connected, then in solving the problem in this latter case. This strategy was proved efficient for representation-finite algebras (that is, algebras having only finitely many isomorphism classes of indecomposable modules) and representation-finite simply connected algebras are by now well-understood: see, for instance [5], [7],[8]. While little is known about covering techniques in the representation-infinite case, it is clearly an interesting problem to describe the representation-infinite simply connected algebras. The objective of this paper is to give a criterion for the simple connectedness of a class of (mostly representationinfinite) algebras.

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 39 , Issue 1 , January 1997, pp. 21 - 27
Copyright
Copyright © Glasgow Mathematical Journal Trust 1997

References

REFERENCES

1.Assem, I. and Skowroński, A., On some classes of simply connected algebras, Proc. London Math. Soc. (3) 56 (1988), 417–450.CrossRefGoogle Scholar
2.Assem, I. and Peña, J. A. de la, The fundamental groups of a triangular algebra, Comm. Algebra 24 (1) (1996), 187–208.CrossRefGoogle Scholar
3.Auslander, M., Representation dimension of artin algebras, Queen Mary College Mathematics Notes (1971), London.Google Scholar
4.Auslander, M., Reiten, I. and Smalø, S., Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics 36 (Cambridge University Press, 1994).Google Scholar
5.Bautista, R., Larrion, F., and Salmeron, L., On simply connected algebras, J. London Math. Soc. (2) 27 (1983), 212–220.CrossRefGoogle Scholar
6.Bongartz, K., A criterion for finite representation type, Math. Ann. 269 (1984), 1–12.CrossRefGoogle Scholar
7.Bongartz, K. and Gabriel, P., Covering spaces in representation-theory, Invent. Math. 65 (1982), 331–378.CrossRefGoogle Scholar
8.Bretscher, O. and Gabriel, P., The standard form of a representation-finite algebra, Bull. Soc. Math. France 111 (1983), 21–40.CrossRefGoogle Scholar
9.Cartan, H. and Eilenberg, S., Homological algebra (Princeton University Press, 1956).Google Scholar
10.Happel, D., Hochschild cohomology of finite-dimensional algebras, Séminaire d'Algèbre P. Dubreil et M.-P. Malliavin (Paris, 1987–1988), Lecture Notes in Mathematics 1404 (Springer, 1989), 108–126.Google Scholar
11.Happel, D., Hochschild cohomology of Auslander algebras, Topics in Algebra, Banach Center Publ. 26 part 1 (PWN, 1990), 303–310.Google Scholar
12.Larrión, F., Algebras forestales, Ph.D. Thesis, Universidad Nacional Autónoma de México, 1981.Google Scholar
13.Martínez-Villa, R. and de ta Pena, J. A., The universal cover of a quiver with relations, J. Pure Appl. Algebra 30 (1983), 277–292.CrossRefGoogle Scholar
14.Ringel, C. M., Tame algebras and integral quadratic forms, Lecture Notes in Mathematics, 1099 (Springer, 1984).CrossRefGoogle Scholar
15.Skowrorński, A., Simply connected algebras and Hochschild cohomologies, Proceedings of the Sixth International Conference on Representations of Algebras (Ottawa, 1992), Canad. Math. Soc. Proc. 14 (1993), 431–437.Google Scholar