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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: Given an Artinian algebra $ A$ over a field $ k$, there are several combinatorial objects associated to $ A$. They are the diagram $ D_A$ as defined by Drozd and Kirichenko, the natural quiver $ \Delta _A$ defined by Li (cf. Section 2), and a generalized version of $ k$-species $ (A/r, r/r^2)$ with $ r$ being the Jacobson radical of $ A$. When $ A$ is splitting over the field $ k$, the diagram $ D_A$ and the well-known Ext-quiver $ \Gamma _A$ 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 $ A$.


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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.