(*)-Serial coalgebras

Abstract

In this paper, we introduce the notion of (*)-serial coalgebras which is a generalization of serial coalgebras. We investigate the properties of (*)-serial coalgebras and their comodules, and obtain sufficient and necessary conditions for a basic coalgebra to be (*)-serial.

Keywords

Coalgebra / biserial comodule / (*)-serial coalgebra / quiver

References

[1] Assem I., Simson D., Skowroński A. Elements of the Representation Theory of Associative Algebras, 2006, Cambridge: Cambridge University Press

[2] Auslander M., Reiten I., Smalϕ S. O. Representation Theory of Associative Algebras, 1995, Cambridge: Cambridge University Press

[3] Chen S. W., Huang H., Zhang P. Dual Gabriel theorem with applications. Sci China, Ser A, 2006, 49: 9-26

[4] Chin W. Special biserial coalgebras and representations of quantum SL(2). arXiv: math/0609370V2 [math, QA], 23 Mar 2011

[5] Chin W., Montgomery S. Basic coalgebra. In: Modular Interfaces (Riverside, CA, 1995), 1997, Providence: Amer Math Soc, 41-47

[6] Cuadra J., Gómez-Torrecillas J. Idempotents and Morita-Takeuchi Theory. Comm Algebra, 2002, 30: 2405-2426

[7] Cuadra J., Gómez-Torrecillas J. Serial coalgebras. J Pure Appl Algebra, 2004, 189: 89-107

[8] Doi Y. Homological coalgebra. J Math Soc Japan, 1981, 33(1): 31-50

[9] Drozd Y. A., Kirichenko V. V. Finite Dimensional Algebras, 1994, Berlin: Springer

[10] Gómez-Torrecillas J., Năstăsescu C. Quasi-Frobenius coalgebras. J Algebra, 1995, 174: 909-923

[11] Gómez-Torrecillas J., Navarro G. Serial coalgebras and valued Gabriel quivers. J Algebra, 2008, 319: 5039-5059

[12] Green J. A. Locally finite representations. J Algebra, 1976, 41: 137-171

[13] Kosakowska J., Simson D. Hereditary coalgebras and representations of species. J Algebra, 2005, 293: 457-505

[14] Lin B. I.-Peng. Semiperfect coalgebras. J Algebra, 1977, 49: 357-373

[15] Montgomery S. Hopf Algebras and Their Actions on Rings, 1993, Providence: Amer Math Soc

[16] Montgomery S. Indecomposable coalgebras, simple comodules and pointed Hopf algebras. Proc Amer Math Soc, 1995, 123: 2343-2351

[17] Năstăsescu C., Torrecillas B., Zhang Y. H. Hereditary coalgebras. Comm Algebra, 1996, 24(4): 1521-1528

[18] Nowak S., Simson D. Locally Dynkin quivers and hereditary coalgebras whose left C-comodules are direct sum of finite dimensional comodules. Comm Algebra, 2002, 30(1): 455-476

[19] Simson D. Coalgebras, comodules, pseudocompact algebras and tame comodule type. Colloq Math, 2001, 90: 101-150

[20] Simson D. Happel D., Zhang Y. B. Coalgebras of tame comodule type. Representations of Algebras, Proceedings ICRA-9, II, 2002, Beijing: Beijing Normal Univ Press, 450-486

[21] Simson D. Path coalgebras of quivers with relations and a tame-wild dichotomy problem for coalgebras, 2005, 236: 465-492

[22] Simson D. Irreducible morphisms, the Gabriel-valued quiver and colocalizations for coalgebras. Int J Math Sci, 2006, 72: 1-6

[23] Simson D. Localizing embeddings of comodule categories with applications to tame and Euler coalgebras. J Algebra, 2007, 312: 455-494

[24] Simson D. Skowroński A., Yamata K. Coalgebras of tame comodule type, comodule categories and tame-wild dichotomy problem. Proc ICRA-XIV, Tokyo Aug 2010, 2011, Zürich: European Math Soc Publishing House

[25] Sweedler M. E. Hopf Algebras, 1969, New York: Benjamin

[26] Woodcock D. Some categorical remarks on the representation theory of coalgebras. Comm Algebra, 1997, 25: 2775-2794

[27] Yao H., Fan Weili. Finite dimensional (*)-Serial algebras. Sci China, Ser A, 2010, 53(12): 3049-3056