Approach to Artinian algebras via natural quivers

Authors:
Fang Li and Zongzhu Lin

Journal:
Trans. Amer. Math. Soc. **364** (2012), 1395-1411

MSC (2010):
Primary 16G10, 16G20; Secondary 16P20, 13E10

Published electronically:
November 7, 2011

MathSciNet review:
2869180

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Abstract | References | Similar Articles | Additional Information

Abstract: Given an Artinian algebra over a field , there are several combinatorial objects associated to . They are the diagram as defined by Drozd and Kirichenko, the natural quiver defined by Li (cf. Section 2), and a generalized version of -species with being the Jacobson radical of . When is splitting over the field , the diagram and the well-known Ext-quiver are the same. The main objective of this paper is to investigate the relations among these combinatorial objects and in turn to use these relations to give a characterization of the algebra .

**[ASS]**I. Assem, D. Simson and A. Skowronski,*Elements of the Representation Theory of Associative Algebras Vol I: Techniques of Representation Theory*, London Mathematical Society Student Texts 65, Cambridge University Press, Cambridge, 2006. MR**2197389 (2006j:16020)****[ARS]**M. Auslander, I. Reiten and S. O. Smalø,*Representation Theory of Artin Algebra*, Cambridge University Press, Cambridge, 1995. MR**1314422 (96c:16015)****[B]**K. Bongartz, A geometric version of the Morita equivalence.*J. Algebra***139**(1991), no. 1, 159-171. MR**1106345 (92f:16008)****[CL]**F. U. Coelho and S. X. Liu, Generalized path algebras, pp. 53-66 in*Interactions between ring theory and repersentations of algebras*(Murcia), Lecture Notes in Pure and Appl. Math, 210, Marcel-Dekker, New York, 2000. MR**1758401 (2001c:16027)****[DR]**V. Dlab and C.M. Ringel,*Indecomposable representations of graphs and algebras*, Mem. Amer. Math. Soc.**6**(1976), no. 173, MR**0447344 (56:5657)****[DK]**Y. A. Drozd and V. V. Kirichenko,*Finite Dimensional Algebras*, Springer-Verlag, Berlin, 1994. MR**1284468 (95i:16001)****[HGK]**M. Hazewinkel, N. Gubareni and V. V.Kirichenko,*Algebras, Rings and Modules*, Vol. 1, Mathematics and Its Applications Vol. 575, Kluwer Academic Publishers, New York, 2005. MR**2106764 (2006a:16001)****[KY]**M. Kontsevich and Y. Soibelman, Notes on -algebras, -categories and non-commutative geometry, Homological mirror symmetry, 153-219, Lect. Notes in Phys., 757, Springer, Berlin, 2009. MR**2596638 (2011f:53183)****[Lam]**T. Y. Lam,*A First Course in Noncommutative Rings*, Graduate Texts in Mathematics 131, Springer-Verlag, New York, 1991. MR**1125071 (92f:16001)****[Li]**F. Li, Characterization of left Artinian algebras through pseudo path algebras,*J. Australia Math. Soc.*,**83**(2007), 385-416. MR**2415878 (2009f:16022)****[LC]**F. Li and L. L. Chen, The natural quiver of an Artinian algebra, to appear in*Algebras and Representation Theory*, online, 2010.**[LW]**F. Li and D. W. Wen, Ext-quiver, AR-quiver and natural quiver of an algebra, in*Geometry, Analysis and Topology of Discrete Groups*, Advanced Lectures in Mathematics 6, Editors: Lizhen Ji, Kefeng Liu, Lo Yang, Shing-Tung Yau, Higher Education Press and International Press, Beijing, 2008.**[Liu]**G. X. Liu,*Classification of finite dimensional basic Hopf algebras and related topics*, Doctoral Dissertation, Zhejiang University, China, 2005.**[P]**R. S. Pierce,*Associative Algebras*, Springer-Verlag, New York, 1982. MR**674652 (84c:16001)****[R]**C.M. Ringel, Representations of -species and bimodules,*J. Algebra***41**(1976), no. 2, 269-302. MR**0422350 (54:10340)**

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Additional Information

**Fang Li**

Affiliation:
Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang 310027, People’s Republic of China

Email:
fangli@cms.zju.edu.cn

**Zongzhu Lin**

Affiliation:
Department of Mathematics, Kansas State University, Manhattan, Kansas 66506

Email:
zlin@math.ksu.edu

DOI:
http://dx.doi.org/10.1090/S0002-9947-2011-05410-3

Received by editor(s):
August 14, 2009

Received by editor(s) in revised form:
May 18, 2010

Published electronically:
November 7, 2011

Additional Notes:
This project was supported by the National Natural Science Foundation of China (No. 10871170) and the Natural Science Foundation of Zhejiang Province of China (No. D7080064)

The second author was supported in part by an NSA grant and the NSF I/RD program

Article copyright:
© Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.