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Representation-Directed Diamonds

Published online by Cambridge University Press:  01 February 2010

Peter Dräxler
Peter Dräxler
Affiliation:
SFB 343, Universität Bielefeld, P. O. Box 100131, D-33501 Bielefeld, Germany, draexler@mathematik.uni-bielefeld.de

Abstract

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A module over a finite-dimensional algebra is called a ‘diamond’ if it has a simple top and a simple socle. Using covering theory, the classification of all diamonds for algebras of finite representation type over algebraically closed fields can be reduced to representation-directed algebras. The author proves a criterion referring to the positive roots of the corresponding Tits quadratic form, which makes it easy to check whether a representation-directed algebra has a faithful diamond. Using an implementation of this criterion in the CREP program system on representation theory, he is able to classify all exceptional representation-directed algebras having a faithful diamond. He obtains a list of 157 algebras up to isomorphism and duality. The 52 maximal members of this list are presented at the end of this paper.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 4 , 2001 , pp. 14 - 21
Copyright
Copyright © London Mathematical Society 2001

References

1.Bongartz, K., ‘Treue einfach zusammenhängende Algebren I’, Comment Math. Helv. 57 (1982) 282–330.CrossRefGoogle Scholar
2.Bongartz, K., ‘Algebras and quadratic forms’, J. London Math. Soc. 28 (1983) 461–469.Google Scholar
3.Bongartz, K. and Gabriel, P., Covering spaces in representation theory, Invent. Math. 65 (1982) 331–378.Google Scholar
4.Dräxler, P., ‘Aufrichtige gerichtete Ausnahmealgebren’, Bayreuth. Math. Schr. 29 (1989).Google Scholar
5.Dräxler, P., ‘Sur les algèbres exceptionalles de Bongartz’, C. R. Acad. Sci. Paris Sér I Math. 311 (1990) 495–498.Google Scholar
6.Dräxler, P. and Nörenberg, R., ‘Classification problems in the combinatorial representation theory of finite-dimensional algebras’, Computational methods for representations of groups and algebras, Progr. Math. 73 (Birkhaüser, Boston, MA, 1999) 3–28.Google Scholar
7.Gabriel, P., ‘Unzerlegbare Darstellungen I’, Manuscr. Math. 6 (1972) 71–103.Google Scholar
8.Gabriel, P., Auslander-Reiten sequences and representation-finite algebras, Lecture Notes in Math. 831 (Springer, New York, 1980) 1–71.Google Scholar
9.Gabriel, P. and Roiter, A. V., ‘Representations of finite-dimensional algebras', Encyclopedia Math. Sci. Vol. 73, Algebra VIII (Ed.Kostrikin, A. I. and Shafarevich, I. V., Springer, Berlin/Heidelberg/New York, 1992).Google Scholar
10.Ringel, C. M., Tame algebras and integral quadratic forms, Lecture Notes in Math. 1099 (Springer, New York, 1984).CrossRefGoogle Scholar
11.Ringel, C. M., ‘Infinite length modules. Some examples as introduction', Infinite length modules, Trends in Mathematics (Ed.Krause, H. and Ringelin, C. M., Birkhäuser, Basel/Boston/Berlin, 2000) 1–73.Google Scholar