(*)-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.

Key words： Coalgebra; biserial comodule; (*)-serial coalgebra; quiver

References

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

2 Auslander M, Reiten I, Smalϕ S O. Representation Theory of Associative Algebras. Cambridge Studies in Advanced Mathematics, 36. Cambridge: Cambridge University Press, 1995

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) AMS/IP Stud Adv Math, Vol 4. Providence: Amer Math Soc, 1997, 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. Berlin: Springer, 1994

10 Gómez-Torrecillas J, Nisăstăsescu 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. CBMS, 82. Providence: Amer Math Soc, 1993

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

21 Simson D. Path coalgebras of quivers with relations and a tame-wild dichotomy problem for coalgebras. Lecture Notes in Pure and Applied Mathematics, 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. Coalgebras of tame comodule type, comodule categories and tame-wild dichotomy problem. In: Skowroński A, Yamata K, eds. Proc ICRA-XIV, Tokyo Aug 2010. Series of Congress Reports. Zürich: European Math Soc Publishing House, 2011

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

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

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