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ALGEBRAS OF GENERALIZED DIHEDRAL TYPE

Part of: Representation theory of groups Representation theory of rings and algebras Modules, bimodules and ideals

Published online by Cambridge University Press:  26 February 2019

KARIN ERDMANN  and
ANDRZEJ SKOWROŃSKI
KARIN ERDMANN
Affiliation:
Mathematical Institute, Oxford University, ROQ, Oxford OX2 6GG, UK email erdmann@maths.ox.ac.uk
ANDRZEJ SKOWROŃSKI
Affiliation:
Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland email skowron@mat.uni.torun.pl
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Abstract

We provide a complete classification of all algebras of generalized dihedral type, which are natural generalizations of algebras which occurred in the study of blocks of group algebras with dihedral defect groups. This gives a description by quivers and relations coming from surface triangulations.

MSC classification

Primary: 16D50: Injective modules, self-injective rings 16G20: Representations of quivers and partially ordered sets 16G60: Representation type (finite, tame, wild, etc.) 16G70: Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers 20C20: Modular representations and characters
Type
Article
Information
Nagoya Mathematical Journal , Volume 240 , December 2020 , pp. 181 - 236
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Copyright
© 2019 Foundation Nagoya Mathematical Journal

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Footnotes

The authors gratefully acknowledge support from the research grant DEC-2011/02/A/ST1/00216 of the National Science Center Poland.

References

Assem, I., Simson, D. and Skowroński, A., Elements of the Representation Theory of Associative Algebras 1: Techniques of Representation Theory, London Mathematical Society Student Texts 65, Cambridge University Press, Cambridge, 2006.10.1017/CBO9780511614309Google Scholar
Białkowski, J., Erdmann, K. and Skowroński, A., Deformed preprojective algebras of generalized Dynkin type, Trans. Amer. Math. Soc. 359 (2007), 2625–2650.10.1090/S0002-9947-07-03948-7Google Scholar
Białkowski, J., Erdmann, K. and Skowroński, A., Periodicity of self-injective algebras of polynomial growth, J. Algebra 443 (2015), 200–269.10.1016/j.jalgebra.2015.05.033Google Scholar
Białkowski, J. and Skowroński, A., On tame weakly symmetric algebras having only periodic modules, Arch. Math. (Basel) 81 (2003), 142–154.10.1007/s00013-003-0816-yGoogle Scholar
Bondarenko, V. M. and Drozd, Y. A., The representation type of finite groups, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. 71 (1977), 24–41.Google Scholar
Brauer, R., On 2-blocks with dihedral defect groups, Sympos. Math. 13 (1974), 367–393.Google Scholar
Butler, M. C. R. and Ringel, C. M., Auslander–Reiten sequences with few middle terms and applications to string algebras, Comm. Algebra 15 (1987), 145–179.10.1080/00927878708823416Google Scholar
Caldero, P., Chapoton, F. and Schiffler, R., Quivers with relations arising from clusters (A n case), Trans. Amer. Math. Soc. 358 (2006), 1347–1364.10.1090/S0002-9947-05-03753-0Google Scholar
Carlson, S. C., Topology of Surfaces, Knots and Manifolds, A first Undergraduate Course, John Wiley & Sons, New York, 2001.Google Scholar
Crawley-Boevey, W., On tame algebras and bocses, Proc. Lond. Math. Soc. 56 (1988), 451–483.10.1112/plms/s3-56.3.451Google Scholar
Crawley-Boevey, W., Tameness of biserial algebras, Arch. Math. (Basel) 65 (1995), 399–407.10.1007/BF01198070Google Scholar
Derksen, H., Weyman, J. and Zelevinsky, A., Quivers with potentials and their representations. I. Mutations, Selecta Math. (N.S.) 14 (2008), 59–119.10.1007/s00029-008-0057-9Google Scholar
Donovan, P. W., Dihedral defect groups, J. Algebra 56 (1979), 184–206.10.1016/0021-8693(79)90332-6Google Scholar
Donovan, P. W. and Freislich, M.-R., The indecomposable modular representations of certain groups with dihedral Sylow subgroup, Math. Ann. 238 (1978), 207–216.10.1007/BF01420248Google Scholar
Dowbor, P. and Skowroński, A., On Galois coverings of tame algebras, Arch. Math. (Basel) 44 (1985), 522–529.10.1007/BF01193992Google Scholar
Dowbor, P. and Skowroński, A., On the representation type of locally bounded categories, Tsukuba J. Math. 10 (1986), 63–72.10.21099/tkbjm/1496160389Google Scholar
Dowbor, P. and Skowroński, A., Galois coverings of representation-infinite algebras, Comment. Math. Helv. 62 (1987), 311–337.10.1007/BF02564450Google Scholar
Drozd, Y. A., “Tame and wild matrix problems”, in Representation Theory II, Lecture Notes in Mathematics 832, Springer, Berlin–Heidelberg–New York, 1980, 242–258.10.1007/BFb0088467Google Scholar
Erdmann, K., Principal blocks of groups with dihedral Sylow 2-subgroups, Comm. Algebra 5 (1977), 665–694.10.1080/00927877708822186Google Scholar
Erdmann, K., Blocks whose defect groups are Klein four groups: a correction, J. Algebra 76 (1982), 505–518.10.1016/0021-8693(82)90228-9Google Scholar
Erdmann, K., Algebras and dihedral defect groups, Proc. Lond. Math. Soc. 54 (1987), 88–114.10.1112/plms/s3-54.1.88Google Scholar
Erdmann, K., On the local structure of tame blocks, Astérisque 181–182 (1990), 173–189.Google Scholar
Erdmann, K., Blocks of Tame Representation Type and Related Algebras, Lecture Notes in Mathematics 1428, Springer, Berlin–Heidelberg–New York, 1990.10.1007/BFb0084003Google Scholar
Erdmann, K. and Michler, G. O., Blocks with dihedral defect groups in solvable groups, Math. Z. 154 (1977), 143–151.10.1007/BF01241827Google Scholar
Erdmann, K. and Skowroński, A., On Auslander–Reiten components of blocks and self-injective biserial algebras, Trans. Amer. Math. Soc. 330 (1992), 165–189.10.1090/S0002-9947-1992-1144759-7Google Scholar
Erdmann, K. and Skowroński, A., The stable Calabi–Yau dimension of tame symmetric algebras, J. Math. Soc. Japan 58 (2006), 97–128.10.2969/jmsj/1145287095Google Scholar
Erdmann, K. and Skowroński, A., “Periodic algebras”, in Representations of Algebras and Related Topics, European Mathematical Society Series Congress Reports, Zürich, 2008, 201–251.Google Scholar
Erdmann, K. and Skowroński, A., Weighted surface algebras, J. Algebra 505 (2018), 490–558.10.1016/j.jalgebra.2018.02.033Google Scholar
Erdmann, K. and Skowroński, A., From Brauer graph algebras to biserial weighted surface algebras, J. Algebr. Combin. (2018), in press; doi:10.1007/s10801-018-0867-6.Google Scholar
Erdmann, K. and Skowroński, A., Algebras of generalized quaternion type, Preprint, 2017, arXiv:abs/1710.09640.Google Scholar
Erdmann, K. and Snashall, N., “Preprojective algebras of Dynkin type, periodicity and the second Hochschild cohomology”, in Algebras and Modules II, CMS Conference Proceedings 24, American Mathematical Society, Providence, RI, 1998, 183–193.Google Scholar
Fock, V. and Goncharov, A., Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. IHES 103 (2006), 1–211.10.1007/s10240-006-0039-4Google Scholar
Fomin, S., Shapiro, M. and Thurston, D., Cluster algebras and triangulated surfaces. Part I. Cluster complexes, Acta Math. 201 (2008), 83–146.10.1007/s11511-008-0030-7Google Scholar
Fomin, S. and Zelevinsky, A., Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), 497–529.10.1090/S0894-0347-01-00385-XGoogle Scholar
Fuller, K. R., “Biserial rings”, in Ring Theory, Lecture Notes in Mathematics 734, Springer, Berlin, 1979, 64–90.Google Scholar
Geiss, C., On degenerations of tame and wild algebras, Arch. Math. (Basel) 64 (1995), 11–16.10.1007/BF01193544Google Scholar
Gekhtman, M., Shapiro, M. and Vainshtein, A., Cluster algebras and Poisson geometry, Moscow Math. J. 3 (2003), 899–934.10.17323/1609-4514-2003-3-3-899-934Google Scholar
Happel, D., Triangulated Categories in the Representation Theory of Finite-Dimensional Algebras, London Mathematical Society Lecture Note Series 119, Cambridge University Press, Cambridge, 1988.10.1017/CBO9780511629228Google Scholar
Hatcher, A., Algebraic Topology, Cambridge University Press, Cambridge, 2002.Google Scholar
Katok, A. and Climenhaga, V., Lectures on Surfaces. (Almost) Everything You Wanted to Know About Them, Student Math. Library 46, American Mathematical Society, Providence, RI, 2008.Google Scholar
Kinsey, L. C., Topology of Surfaces, Undergraduate Texts in Mathematics, Springer, New York, 1993.10.1007/978-1-4612-0899-0Google Scholar
Kraft, H., “Geometric methods in representation theory”, in Representations of Algebras, Lecture Notes in Mathematics 944, Springer, Berlin–Heidelberg–New York, 1982, 180–258.10.1007/BFb0094059Google Scholar
Külshammer, B., Lectures on Block Theory, London Mathematical Society Lecture Note Series 161, Cambridge University Press, Cambridge, 1991.10.1017/CBO9780511565786Google Scholar
Labardini-Fragoso, D., Quivers with potentials associated to triangulated surfaces, Proc. Lond. Math. Soc. 98 (2009), 797–839.10.1112/plms/pdn051Google Scholar
Ladkani, S., On Jacobian algebras from closed surfaces, Preprint, 2012, arXiv:abs/1207.3778.Google Scholar
Ladkani, S., Algebras of quasi-quaternion type, Preprint, 2014, arXiv:abs/1404.6834.Google Scholar
Lampe, P., Diophantine equations via cluster transformations, J. Algebra 462 (2016), 320–337.10.1016/j.jalgebra.2016.04.033Google Scholar
Landrock, P., The principal block of finite groups with dihedral Sylow 2-subgroups, J. Algebra 39 (1976), 410–428.10.1016/0021-8693(76)90046-6Google Scholar
Markoff, A., Sur les formes quadratiques binaires indéfinies, Math. Ann. 15(3–4) (1879), 381–406.10.1007/BF02086269Google Scholar
Pogorzały, Z. and Skowroński, A., Selfinjective biserial standard algebras, J. Algebra 138 (1991), 491–504.10.1016/0021-8693(91)90183-9Google Scholar
Ringel, C. M., The indecomposable representations of the dihedral 2-groups, Math. Ann. 214 (1975), 19–34.10.1007/BF01428252Google Scholar
Simson, D. and Skowroński, A., Elements of the Representation Theory of Associative Algebras 3: Representation-Infinite Tilted Algebras, London Mathematical Society Student Texts 72, Cambridge University Press, Cambridge, 2007.Google Scholar
Skowroński, A., Group algebras of polynomial growth, Manuscripta Math. 59 (1987), 499–516.10.1007/BF01170851Google Scholar
Skowroński, A., Selfinjective algebras of polynomial growth, Math. Ann. 285 (1989), 177–199.10.1007/BF01443513Google Scholar
Skowroński, A., Selfinjective algebras: finite and tame type, Contemp. Math. 406 (2006), 169–238.10.1090/conm/406/07658Google Scholar
Skowroński, A. and Waschbüsch, J., Representation-finite biserial algebras, J. Reine Angew. Math. 345 (1983), 172–181.Google Scholar
Skowroński, A. and Yamagata, K., Frobenius Algebras I. Basic Representation Theory, European Mathematical Society Textbooks in Mathematics, European Mathematical Society, Zürich, 2011.10.4171/102Google Scholar
Vila-Freyer, R. and Crawley-Boevey, W., The structure of biserial algebras, J. London Math. Soc. 57 (1998), 41–54.10.1112/S0024610798005821Google Scholar
Wald, B. and Waschbüsch, J., Tame biserial algebras, J. Algebra 95 (1985), 480–500.10.1016/0021-8693(85)90119-XGoogle Scholar