DERIVED EQUIVALENCE CLASSIFICATION OF SYMMETRIC ALGEBRAS OF POLYNOMIAL GROWTH

HOLM, THORSTEN ; SKOWROŃSKI, ANDRZEJ

Glasgow mathematical journal, Tome 53 (2011) no. 2, pp. 277-291

Résumé

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.

Détail

DOI : 10.1017/S0017089510000698

Mots-clés : 16G10, 18E30, 16D50, 16G60

Bibliographie

[1] 1.Al-Kadi, D., Distinguishing derived equivalence classes using the second Hochschild cohomology group, Colloq. Math. (to appear). Google Scholar

[2] 2.Al-Nofayee, S., Derived equivalence for selfinjective algebras and t-structures, PhD Thesis (Bristol University, 2004). Google Scholar

[3] 3.Asashiba, H., The derived equivalence classification of representation-finite selfinjective algebras, J. Algebra 214 (1999), 182–221. Google Scholar | DOI

[4] 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). Google Scholar | DOI

[5] 5.Białkowski, J., Erdmann, K. and Skowroński, A., Deformed preprojective algebras of generalized Dynkin type, Trans. Amer. Math. Soc. 359 (2007), 2625–2650. Google Scholar | DOI

[6] 6.Białkowski, J., Holm, T. and Skowroński, A., Derived equivalences for tame weakly symmetric algebras having only periodic modules, J. Algebra 269 (2003), 652–668. Google Scholar | DOI

[7] 7.Białkowski, J., Holm, T. and Skowroński, A., On nonstandard tame selfinjective algebras having only periodic modules, Colloq. Math. 97 (2003), 33–47. Google Scholar | DOI

[8] 8.Białkowski, J. and Skowroński, A., On tame weakly symmetric algebras algebras having only periodic modules, Arch. Math. 81 (2003), 142–154. Google Scholar | DOI

[9] 9.Białkowski, J. and Skowroński, A., Socle deformations of selfinjective algebras of tubular type, J. Math. Soc. Japan 56 (2004), 687–716. Google Scholar | DOI

[10] 10.Bocian, R., Holm, T. and Skowroński, A., Derived equivalence classification of weakly symmetric algebras of Euclidean type, J. Pure Appl. Algebra 191 (2004), 43–74. Google Scholar | DOI

[11] 11.Bocian, R., Holm, T. and Skowroński, A., Derived equivalence classification of nonstandard algebras of domestic type, Comm. Alg. 35 (2007), 515–526. Google Scholar | DOI

[12] 12.Drozd, Y. A., Tame and wild matrix problems, in Representation Theory II, (Lecture Notes in Mathematics vol. 832) (Springer, Berlin, 1980), 242–258. Google Scholar | DOI

[13] 13.Erdmann, K., Blocks of Tame Representation Type and Related Algebras, (Lecture Notes in Mathematics vol. 1428) (Springer: Berlin, 1990). Google Scholar | DOI

[14] 14.Erdmann, K. and Skowroński, A., Classification of tame symmetric algebras with periodic modules (in preparation). Google Scholar

[15] 15.Erdmann, K. and Skowroński, A., The stable Calabi-Yau dimension of tame symmetric algebras, J. Math. Soc. Japan 58 (2006), 97–128. Google Scholar | DOI

[16] 16.Gerstenhaber, M., The cohomology structure of an associative ring, Ann. Math. 78 (2), (1963), 267–288. Google Scholar | DOI

[17] 17.Happel, D., On the derived category of a finite-dimensional algebra, Comment. Math. Helv. 62 (1987), 339–389. Google Scholar | DOI

[18] 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). Google Scholar | DOI

[19] 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), 156–180. Google Scholar | DOI

[20] 20.Héthelyi, L., Horváth, E., Külshammer, B. and Murray, J., Central ideals and Cartan invariants of symmetric algebras, J. Algebra 293 (2005), 243–260. Google Scholar | DOI

[21] 21.Holm, T., Derived equivalence classification of algebras of dihedral, semidihedral, and quaternion type, J. Algebra 211 (1999), 159–205. Google Scholar | DOI

[22] 22.Holm, T. and Skowroński, A., Derived equivalence classification of symmetric algebras of domestic type, J. Math. Soc. Japan 58 (2006), 1133–1149. Google Scholar | DOI

[23] 23.Holm, T. and Zimmermann, A., Generalized Reynolds ideals and derived equivalences for algebras of dihedral and semidihedral type, J. Algebra 320 (2008), 3425–3437. Google Scholar | DOI

[24] 24.Krause, H. and Zwara, G., Stable equivalence and generic modules, Bull. London Math. Soc. 32 (2000), 615–618. Google Scholar | DOI

[25] 25.Külshammer, B., Bemerkungen über die Gruppenalgebra als symmetrische Algebra I, J. Algebra 72 (1981), 1–7. Google Scholar | DOI

[26] 26.Külshammer, B., Bemerkungen über die Gruppenalgebra als symmetrische Algebra II, J. Algebra 75 (1982), 59–69. Google Scholar | DOI

[27] 27.Külshammer, B., Bemerkungen über die Gruppenalgebra als symmetrische Algebra III, J. Algebra 88 (1984), 279–291. Google Scholar | DOI

[28] 28.Külshammer, B., Bemerkungen über die Gruppenalgebra als symmetrische Algebra IV, J. Algebra 93 (1985), 310–323. Google Scholar | DOI

[29] 29.Külshammer, B., Group-theoretical descriptions of ring theoretical invariants of group algebras, Prog. Math. 95 (1991), 425–441. Google Scholar

[30] 30.Nakayama, T., On Frobeniusean algebras I, Ann. Math. 40 (1939), 611–633. Google Scholar | DOI

[31] 31.Nakayama, T., On Frobeniusean algebras II, Ann. of Math. 42 (1941), 1–21. Google Scholar | DOI

[32] 32.Nehring, J. and Skowroński, A., Polynomial growth trivial extensions of simply connected algebras, Fund. Math. 132 (1989), 117–134. Google Scholar | DOI

[33] 33.Rickard, J., Morita theory for derived categories, J. London Math. Soc. 39 (1989), 436–456. Google Scholar | DOI

[34] 34.Rickard, J., Derived categories and stable equivalence, J. Pure Appl. Algebra 61 (1989), 303–317. Google Scholar | DOI

[35] 35.Rickard, J., Derived equivalences as derived functors, J. London Math. Soc. 43 (1991), 37–48. Google Scholar | DOI

[36] 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] 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] 38.Skowroński, A., Classification of self-injective algebras of polynomial growth (in preparation). Google Scholar

[39] 39.Skowroński, A., Selfinjective algebras of polynomial growth, Math. Ann. 285 (1989), 177–199. Google Scholar | DOI

[40] 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), 169–238. Google Scholar | DOI

[41] 41.Yamagata, K., Frobenius algebras, in Handbook of Algebra, vol. 1 (Elsevier Science B.V., Amsterdam, 1996), 841–887. Google Scholar | DOI

[42] 42.Zimmermann, A., Invariance of generalized Reynolds ideals under derived equivalences, Math. Proc. R. Ir. Acad. 107 (2007), 1–9. Google Scholar | DOI