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Forbidden Subcategories of Non-Polynomial Growth Tame Simply Connected Algebras

Published online by Cambridge University Press:  20 November 2018

J. A. De La Peña  and
A. Skowroński
J. A. De La Peña
Affiliation:
Instituto de Matemáticas, UNAM Ciudad Universitaria México 04510 D.F. México
A. Skowroński
Affiliation:
Faculty of Mathematics and Informatics Nicholas Copernicus University Chopina 12/18 87-100 Toruń Poland
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Abstract

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Let k be an algebraically closed field and A = kQ/I be a basic finite dimensional k-algebra such that Q is a connected quiver without oriented cycles. Assume that A is strongly simply connected, that is, for every convex subcategory B of A the first Hochschild cohomology H1(B, B) vanishes. The algebra A is sincere if it admits an indecomposable module having all simples as composition factors. We study the structure of strongly simply connected sincere algebras of tame representation type. We show that a sincere, tame, strongly connected algebra A which contains a convex subcategory which is either representation-infinite tilted of type Ẽp, p = 6,7,8, or a tubular algebra, is of polynomial growth.

Keywords

16G1016G60
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 48 , Issue 5 , 01 October 1996 , pp. 1018 - 1043
Copyright
Copyright © Canadian Mathematical Society 1996

References

1. Assem, I. and Skowro, A.ński, On some classes of simply connected algebras, Proc. London Math. Soc., (3) 56(1988), 417–450.Google Scholar
2. Assem, I., Indecomposable modules over multicoil algebras, Math., Scand. 71(1992), 31–61.Google Scholar
3. Assem, I., Multicoil algebras.In: Proc. ICRA VI, Ottawa, 1992. CMS Conf. Proc. 14, Amer. Math. Soc, 1993. 29–68.Google Scholar
4. Assem, I., Skowro, A.ński and Tomé, B., Coil enlargements of algebras, Tsukuba J., Math. 19(1995), 453–479.Google Scholar
5. Bautista, R., Gabriel, P., P., Roiter, A. and Salmeron, L., Representation-finite algebras and multiplicative bases, Invent., Math. 81(1985), 217–285.Google Scholar
6. Bautista, R., Larrion, F. and Salmerón, L., On simply connected algebras, J. London Math. Soc., (2) 27(1983), 212–220.Google Scholar
7. Bongartz, K., and Gabriel, P., Covering spaces in representation theory, Invent., Math. 65(1982), 331–378.Google Scholar
8. Dlab, V. and Ringel, C.M., Indecomposable representations of graphs and algebras, Mem. Amer. Math., Soc. 173(1976).Google Scholar
9. Dowbor, P. and Skowroński, A., Galois coverings of representation-infinite algebras, Comment. Math., Helv. 62(1987), 311–337.Google Scholar
10. Gabriel, P., Auslander-Reiten sequences and representation-finite algebras, Proc. ICRA II, Ottawa, 1979. Lecture Notes in Math. 903, Springer, 1981. 68–105.Google Scholar
11. Geiss, Ch., Tame distributive 2-point algebras. In: Proc. ICRA VI, Ottawa, 1992. CMS Conf. Proc. 14, Amer. Math. Soc, 1993.Google Scholar
12. Kerner, O., Tilting wild algebras, J. London Math., Soc. 39(1989), 29–47.Google Scholar
13. Nehring, J. and Skowroński, A., Polynomial growth trivial extensions of simply connected algebras, Fund., Math. 132(1989), 117–134.Google Scholar
14. de la Peña, J.A., On the dimension of module varieties of tame and wild algebras, Comm., Alg. 19(1991), 1795–1807.Google Scholar
15. de la Peña, J.A., Tame algebras with a sincere directing module, J., Algebra 161(1993), 171–185.Google Scholar
16. de la Peña, J.A., The families ofl-parametric domestic algebras with a sincere directing module.In: Proc ICRA VI, Ottawa, 1992. CMS Conf. Proc 14, Amer. Math. Soc, 1993. 361–392.Google Scholar
17. Ringel, C.M., Tame algebras. In: Lecture Notes in Math. 831, Springer, 1980. 137–287.Google Scholar
18. Ringel, C.M., Tame algebras and integral quadratic forms, Lecture Notes in Math. 1099, Springer, 1984.Google Scholar
19. Simson, D., Linear representations of partially ordered sets and vector space categories, Algebra Logic Appl. 4, Gordon and Breach, 1992.Google Scholar
20. Skowroriski, A., Selfinjective algebras of polynomial growth, Math., Ann. 285(1988), 177–199.Google Scholar
21. Skowroriski, A., Simply connected algebras and Hochschild cohomologies. In: Proc. ICRA VI, Ottawa, 1992. CMS Conf. Proc. 14, Amer. Math. Soc, 1993. 431–447.Google Scholar
22. Skowroriski, A., Cycle-finite algebras, J. Pure Appl., Algebra 103(1995), 105–116.Google Scholar
23. Skowroriski, A., Tame algebras with simply connected Galois coverings, in preparation.Google Scholar
24. Strauss, H., The perpendicular category of a partial tilting module, J., Algebra 144(1991), 43–66.Google Scholar