Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-25T07:03:29.457Z Has data issue: false hasContentIssue false

DERIVED EQUIVALENCE CLASSIFICATION OF SYMMETRIC ALGEBRAS OF POLYNOMIAL GROWTH

Published online by Cambridge University Press:  08 December 2010

THORSTEN HOLM
Affiliation:
Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Fakultät für Mathematik und Physik, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany e-mail: holm@math.uni-hannover.dehttp://www.iazd.uni-hannover.de/~tholm
ANDRZEJ SKOWROŃSKI
Affiliation:
Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland e-mail: skowron@mat.uni.torun.pl
Rights & Permissions [Opens in a new window]

Abstract

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.

We complete the derived equivalence classification of all symmetric algebras of polynomial growth, by solving the subtle problem of distinguishing the standard and nonstandard nondomestic symmetric algebras of polynomial growth up to derived equivalence.

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 2010

References

REFERENCES

1.Al-Kadi, D., Distinguishing derived equivalence classes using the second Hochschild cohomology group, Colloq. Math. (to appear).Google Scholar
2.Al-Nofayee, S., Derived equivalence for selfinjective algebras and t-structures, PhD Thesis (Bristol University, 2004).Google Scholar
3.Asashiba, H., The derived equivalence classification of representation-finite selfinjective algebras, J. Algebra 214 (1999), 182221.CrossRefGoogle Scholar
4.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 vol. 65) (Cambridge University Press, Cambridge, 2006).CrossRefGoogle Scholar
5.Białkowski, J., Erdmann, K. and Skowroński, A., Deformed preprojective algebras of generalized Dynkin type, Trans. Amer. Math. Soc. 359 (2007), 26252650.CrossRefGoogle Scholar
6.Białkowski, J., Holm, T. and Skowroński, A., Derived equivalences for tame weakly symmetric algebras having only periodic modules, J. Algebra 269 (2003), 652668.CrossRefGoogle Scholar
7.Białkowski, J., Holm, T. and Skowroński, A., On nonstandard tame selfinjective algebras having only periodic modules, Colloq. Math. 97 (2003), 3347.CrossRefGoogle Scholar
8.Białkowski, J. and Skowroński, A., On tame weakly symmetric algebras algebras having only periodic modules, Arch. Math. 81 (2003), 142154.CrossRefGoogle Scholar
9.Białkowski, J. and Skowroński, A., Socle deformations of selfinjective algebras of tubular type, J. Math. Soc. Japan 56 (2004), 687716.CrossRefGoogle Scholar
10.Bocian, R., Holm, T. and Skowroński, A., Derived equivalence classification of weakly symmetric algebras of Euclidean type, J. Pure Appl. Algebra 191 (2004), 4374.CrossRefGoogle Scholar
11.Bocian, R., Holm, T. and Skowroński, A., Derived equivalence classification of nonstandard algebras of domestic type, Comm. Alg. 35 (2007), 515526.CrossRefGoogle Scholar
12.Drozd, Y. A., Tame and wild matrix problems, in Representation Theory II, (Lecture Notes in Mathematics vol. 832) (Springer, Berlin, 1980), 242258.CrossRefGoogle Scholar
13.Erdmann, K., Blocks of Tame Representation Type and Related Algebras, (Lecture Notes in Mathematics vol. 1428) (Springer: Berlin, 1990).CrossRefGoogle Scholar
14.Erdmann, K. and Skowroński, A., Classification of tame symmetric algebras with periodic modules (in preparation).Google Scholar
15.Erdmann, K. and Skowroński, A., The stable Calabi-Yau dimension of tame symmetric algebras, J. Math. Soc. Japan 58 (2006), 97128.CrossRefGoogle Scholar
16.Gerstenhaber, M., The cohomology structure of an associative ring, Ann. Math. 78 (2), (1963), 267288.CrossRefGoogle Scholar
17.Happel, D., On the derived category of a finite-dimensional algebra, Comment. Math. Helv. 62 (1987), 339389.CrossRefGoogle Scholar
18.Happel, D., Triangulated Categories in the Representation Theory of Finite Dimensional Algebras, London Math. Soc. (Lecture Note Series vol. 119) (Cambridge University Press, Cambridge, 1988).CrossRefGoogle Scholar
19.Happel, D. and Ringel, C. M., The derived category of a tubular algebra, in Representation Theory I. Finite Dimensional Algebras, (Lecture Notes in Mathematics vol. 1177) (Springer: Berlin, 1986), 156180.CrossRefGoogle Scholar
20.Héthelyi, L., Horváth, E., Külshammer, B. and Murray, J., Central ideals and Cartan invariants of symmetric algebras, J. Algebra 293 (2005), 243260.CrossRefGoogle Scholar
21.Holm, T., Derived equivalence classification of algebras of dihedral, semidihedral, and quaternion type, J. Algebra 211 (1999), 159205.CrossRefGoogle Scholar
22.Holm, T. and Skowroński, A., Derived equivalence classification of symmetric algebras of domestic type, J. Math. Soc. Japan 58 (2006), 11331149.CrossRefGoogle Scholar
23.Holm, T. and Zimmermann, A., Generalized Reynolds ideals and derived equivalences for algebras of dihedral and semidihedral type, J. Algebra 320 (2008), 34253437.CrossRefGoogle Scholar
24.Krause, H. and Zwara, G., Stable equivalence and generic modules, Bull. London Math. Soc. 32 (2000), 615618.CrossRefGoogle Scholar
25.Külshammer, B., Bemerkungen über die Gruppenalgebra als symmetrische Algebra I, J. Algebra 72 (1981), 17.CrossRefGoogle Scholar
26.Külshammer, B., Bemerkungen über die Gruppenalgebra als symmetrische Algebra II, J. Algebra 75 (1982), 5969.CrossRefGoogle Scholar
27.Külshammer, B., Bemerkungen über die Gruppenalgebra als symmetrische Algebra III, J. Algebra 88 (1984), 279291.CrossRefGoogle Scholar
28.Külshammer, B., Bemerkungen über die Gruppenalgebra als symmetrische Algebra IV, J. Algebra 93 (1985), 310323.CrossRefGoogle Scholar
29.Külshammer, B., Group-theoretical descriptions of ring theoretical invariants of group algebras, Prog. Math. 95 (1991), 425441.Google Scholar
30.Nakayama, T., On Frobeniusean algebras I, Ann. Math. 40 (1939), 611633.CrossRefGoogle Scholar
31.Nakayama, T., On Frobeniusean algebras II, Ann. of Math. 42 (1941), 121.CrossRefGoogle Scholar
32.Nehring, J. and Skowroński, A., Polynomial growth trivial extensions of simply connected algebras, Fund. Math. 132 (1989), 117134.CrossRefGoogle Scholar
33.Rickard, J., Morita theory for derived categories, J. London Math. Soc. 39 (1989), 436456.CrossRefGoogle Scholar
34.Rickard, J., Derived categories and stable equivalence, J. Pure Appl. Algebra 61 (1989), 303317.CrossRefGoogle Scholar
35.Rickard, J., Derived equivalences as derived functors, J. London Math. Soc. 43 (1991), 3748.CrossRefGoogle Scholar
36.Simson, D. and 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).Google Scholar
37.Simson, D. and 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).Google Scholar
38.Skowroński, A., Classification of self-injective algebras of polynomial growth (in preparation).Google Scholar
39.Skowroński, A., Selfinjective algebras of polynomial growth, Math. Ann. 285 (1989), 177199.CrossRefGoogle Scholar
40.Skowroński, A., Selfinjective algebras: finite and tame type, in Trends in Representation Theory of Algebras and Related Topics, Contemporary Mathematics, vol. 406, (American Mathematical Society, Providence, RI, 2006), 169238.CrossRefGoogle Scholar
41.Yamagata, K., Frobenius algebras, in Handbook of Algebra, vol. 1 (Elsevier Science B.V., Amsterdam, 1996), 841887.CrossRefGoogle Scholar
42.Zimmermann, A., Invariance of generalized Reynolds ideals under derived equivalences, Math. Proc. R. Ir. Acad. 107 (2007), 19.CrossRefGoogle Scholar