Abstract
We study the strong simple connectedness of finite-dimensional tame algebras over an algebraically closed field, for which the Auslander–Reiten quiver admits a separating family of almost cyclic coherent components. As the main application we describe all analytically rigid algebras in this class.
Article PDF
Similar content being viewed by others
References
Assem, I.: Strongly simply connected derived tubular algebras. In: Representations of Algebras, Lecture Notes in Pure and Appl. Math, vol. 224, pp 21–29. Dekker, New York (2002)
Assem, I., Coelho, F. U., Trepode, S.: Simply connected tame quasi-tilted algebras. J. Pure Appl. Algebra 172, 139–160 (2002)
Assem, I., Liu, S.: Strongly simply connected tilted algebras. Ann. Sci. Math. Québec 21, 13–22 (1997)
Assem, I., Liu, S.: Strongly simply connected algebras. J. Algebra 207, 449–477 (1998)
Assem, I., Liu, S., de la Peña, J. A.: The strong simple connectedness of a tame tilted algebra. Comm. Algebra 28, 1553–1565 (2000)
Assem, I., de la Peña, J. A.: The fundamental groups of a triangular algebra. Comm. Algebra 24, 187–208 (1996)
Assem, I., Simson, D., Skowroński, A.: Elements of the representation theory of associative algebras 1: Techniques of representation theory london mathematical society student texts, vol. 65. Cambridge University Press, Cambridge (2006)
Assem, I., Skowroński, A.: On some classes of simply connected algebras. Proc. London Math. Soc. 56, 417–450 (1988)
Assem, I., Skowroński, A.: Minimal representation-infinite coil algebras. Manuscr. Math. 67, 305–331 (1990)
Assem, I., Skowroński, A.: Multicoil algebras. In: Representations of Algebras. CMS Conf. Proc. 14, Amer. Math. Soc., Providence, RI, pp 29–68 (1993)
Assem, I., Skowroński, A., Tomé, B.: Coil enlargements of algebras. Tsukuba J. Math. 19, 453–479 (1995)
Bautista, R., Larrion, F., Salmeron, L.: On simply connected algebras. J. London. Math. Soc. 27, 212–220 (1983)
Bongartz, K., Gabriel, P.: Covering spaces in representation theory. Invent. Math. 65, 331–378 (1982)
Coelho, F. U., Martins, M. I. R., Tomé, B.: Strongly simply connected coil algebras. Colloq. Math. 99, 91–110 (2004)
Crawley-Boevey, W.: On tame algebras and bocses. Proc. London Math. Soc. 56, 451–483 (1988)
Drozd, Y. A.: Tame and wild matrix problems. In: Representation Theory II, Lecture Notes in Math. 832, pp 242–258. Springer (1980)
Gerstenhaber, M.: On the deformation of rings and algebras. Ann. of Math. 79, 59–103 (1964)
Happel, D., Vossieck, D.: Minimal algebras of infinite representation type with preprojective component. Manuscripta Math. 42, 221–243 (1983)
Hoshino, M.: Modules without self-extensions and Nakayama’s conjecture. Archiv. Math. (Basel) 43, 493–500 (1984)
Kerner, O.: Tilting wild algebras. J. London Math. Soc. 39, 29–47 (1989)
Lenzing, H., Meltzer, H.: Tilting sheaves and concealed-canonical algebras. In: Representation Theory of Algebras, CMS Conference Proc., vol. 18, pp 455–473 (1996)
Lenzing, H., de la Peña, J. A.: Concealed-canonical algebras and separating tubular families. Proc. London Math. Soc. 78, 513–540 (1999)
Lenzing, H., Skowroński, A.: Quasi-tilted algebras of canonical type. Colloq. Math. 71, 161–181 (1996)
Malicki, P.: Generalized coil enlargements of algebras. Colloq. Math. 76, 57–83 (1998)
Malicki, P., de la Peña, J. A., Skowroński, A.: Cycle-finite module categories. In: Algebras, Quivers and Representations - Abel Symposium 2011. Abel Symposia 8, pp 209–252. Springer (2013)
Malicki, P., Skowroński, A.: Almost cyclic coherent components of an Auslander-Reiten quiver. J. Algebra 229, 695–749 (2000)
Malicki, P., Skowroński, A.: Algebras with separating almost cyclic coherent Auslander-Reiten components. J. Algebra 291, 208–237 (2005)
Malicki, P., Skowroński, A.: On the additive categories of generalized standard almost cyclic coherent Auslander-Reiten components. J. Algebra 316, 133–146 (2007)
Malicki, P., Skowroński, A.: Algebras with separating Auslander-Reiten components. In: Representations of Algebras and Related Topics, European Math. Soc. Series Congress Reports, European Math. Soc. Publ. House, Zürich, pp 251–353 (2011)
Malicki, P., Skowroński, A.: Hochschild cohomology of generalized multicoil algebras. Colloq. Math. 136, 231–254 (2014)
Malicki, P., Skowroński, A.: The structure and homological properties of generalized standard Auslander-Reiten components. J. Algebra 518, 1–39 (2019)
Reiten, I., Skowroński, A.: Characterizations of algebras with small homological dimensions. Adv. Math. 179, 122–154 (2003)
Ringel, C. M.: Separating tubular series. In: Séminare d’Algébre Paul Dubreil et Marie-Paul Malliavin, Lecture Notes in Math., vol. 1029, pp 134–158. Springer, Berlin (1983)
Ringel, C. M.: Tame Algebras and Integral Quadratic Forms, p 1984. Springer, Berlin (1099). Lecture Notes in Math.
Simson, D., Skowroński, A.: Elements of the Representation Theory of Associative Algebras 2: Tubes and Concealed Algebras of Euclidean Type London Mathematical Society Student Texts, vol. 71. Cambridge University Press, Cambridge (2007)
Simson, D., Skowroński, A.: Elements of the Representation Theory of Associative Algebras 3: Representation-Infinite Tilted Algebras London Mathematical Society Student Texts, vol. 72. Cambridge University Press, Cambridge (2007)
Skowroński, A.: Algebras of polynomial growth. In: Topics in Algebra. Part 1, PWN Warsaw, vol. 26, pp 535–568 (1990)
Skowroński, A.: Simply connected algebras and Hochschild cohomologies. In: Representations of Algebras, CMS Conference Proc., vol. 14, pp 431–447 (1993)
Skowroński, A.: Generalized standard Auslander-Reiten components. J. Math. Soc. Japan 46, 517–543 (1994)
Skowroński, A.: Cycle-finite algebras. J. Pure Appl. Algebra 103, 105–116 (1995)
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by: Christof Geiss
Dedicated to Ibrahim Assem on the occasion of his 70th birthday.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Malicki, P. The strong simple connectedness of tame algebras with separating almost cyclic coherent Auslander–Reiten components. Algebr Represent Theor 25, 309–340 (2022). https://doi.org/10.1007/s10468-020-10023-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-020-10023-9