Approach to Artinian algebras via natural quivers
Authors:
Fang Li and Zongzhu Lin
Journal:
Trans. Amer. Math. Soc. 364 (2012), 13951411
MSC (2010):
Primary 16G10, 16G20; Secondary 16P20, 13E10
Published electronically:
November 7, 2011
MathSciNet review:
2869180
Fulltext PDF
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 wellknown Extquiver 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]
Ibrahim
Assem, Daniel
Simson, and Andrzej
Skowroński, Elements of the representation theory of
associative algebras. Vol. 1, London Mathematical Society Student
Texts, vol. 65, Cambridge University Press, Cambridge, 2006.
Techniques of representation theory. MR 2197389
(2006j:16020)
 [ARS]
Maurice
Auslander, Idun
Reiten, and SmaløSverre
O., Representation theory of Artin algebras, Cambridge Studies
in Advanced Mathematics, vol. 36, Cambridge University Press,
Cambridge, 1995. MR 1314422
(96c:16015)
 [B]
Klaus
Bongartz, A geometric version of the Morita equivalence, J.
Algebra 139 (1991), no. 1, 159–171. MR 1106345
(92f:16008), http://dx.doi.org/10.1016/00218693(91)90288J
 [CL]
Flávio
Ulhoa Coelho and ShaoXue
Liu, Generalized path algebras, Interactions between ring
theory and representations of algebras (Murcia), Lecture Notes in Pure and
Appl. Math., vol. 210, Dekker, New York, 2000, pp. 53–66.
MR
1758401 (2001c:16027)
 [DR]
Vlastimil
Dlab and Claus
Michael Ringel, Indecomposable representations of graphs and
algebras, Mem. Amer. Math. Soc. 6 (1976),
no. 173, v+57. MR 0447344
(56 #5657)
 [DK]
Yurij
A. Drozd and Vladimir
V. Kirichenko, Finitedimensional algebras, SpringerVerlag,
Berlin, 1994. Translated from the 1980 Russian original and with an
appendix by Vlastimil Dlab. MR 1284468
(95i:16001)
 [HGK]
Michiel
Hazewinkel, Nadiya
Gubareni, and V.
V. Kirichenko, Algebras, rings and modules. Vol. 1,
Mathematics and its Applications, vol. 575, Kluwer Academic
Publishers, Dordrecht, 2004. MR 2106764
(2006a:16001)
 [KY]
M.
Kontsevich and Y.
Soibelman, Notes on 𝐴_{∞}algebras,
𝐴_{∞}categories and noncommutative geometry,
Homological mirror symmetry, Lecture Notes in Phys., vol. 757,
Springer, Berlin, 2009, pp. 153–219. MR 2596638
(2011f:53183)
 [Lam]
T.
Y. Lam, A first course in noncommutative rings, Graduate Texts
in Mathematics, vol. 131, SpringerVerlag, New York, 1991. MR 1125071
(92f:16001)
 [Li]
Fang
Li, Characterization of left Artinian algebras through pseudo path
algebras, J. Aust. Math. Soc. 83 (2007), no. 3,
385–416. MR 2415878
(2009f:16022), http://dx.doi.org/10.1017/S144678870003799X
 [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, Extquiver, ARquiver 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, ShingTung 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]
Richard
S. Pierce, Associative algebras, Graduate Texts in
Mathematics, vol. 88, SpringerVerlag, New York, 1982. Studies in the
History of Modern Science, 9. MR 674652
(84c:16001)
 [R]
Claus
Michael Ringel, Representations of 𝐾species and
bimodules, J. Algebra 41 (1976), no. 2,
269–302. MR 0422350
(54 #10340)
 [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, 159171. MR 1106345 (92f:16008)
 [CL]
 F. U. Coelho and S. X. Liu, Generalized path algebras, pp. 5366 in Interactions between ring theory and repersentations of algebras (Murcia), Lecture Notes in Pure and Appl. Math, 210, MarcelDekker, 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, SpringerVerlag, 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 noncommutative geometry, Homological mirror symmetry, 153219, 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, SpringerVerlag, 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), 385416. 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, Extquiver, ARquiver 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, ShingTung 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, SpringerVerlag, New York, 1982. MR 674652 (84c:16001)
 [R]
 C.M. Ringel, Representations of species and bimodules, J. Algebra 41 (1976), no. 2, 269302. MR 0422350 (54:10340)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC (2010):
16G10,
16G20,
16P20,
13E10
Retrieve articles in all journals
with MSC (2010):
16G10,
16G20,
16P20,
13E10
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/S000299472011054103
PII:
S 00029947(2011)054103
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.
