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Representations as elements in affine composition algebras


Author: Pu Zhang
Journal: Trans. Amer. Math. Soc. 353 (2001), 1221-1249
MSC (2000): Primary 16G20, 17B37; Secondary 16G60, 17B40, 05A30, 18G15
DOI: https://doi.org/10.1090/S0002-9947-00-02613-1
Published electronically: November 16, 2000
MathSciNet review: 1707704
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Abstract: Let $A$ be the path algebra of a Euclidean quiver over a finite field $k$. The aim of this paper is to classify the modules $M$ with the property $[M]\in \mathcal{C}(A)$, where $\mathcal{C}(A)$ is Ringel's composition algebra. Namely, the main result says that if $\vert k\vert \ne 2, 3$, then $[M]\in \mathcal{C}(A)$ if and only if the regular direct summand of $M$ is a direct sum of modules from non-homogeneous tubes with quasi-dimension vectors non-sincere. The main methods are representation theory of affine quivers, the structure of triangular decompositions of tame composition algebras, and the invariant subspaces of skew derivations. As an application, we see that $\mathcal{C}(A) = \mathcal{H}(A)$ if and only if the quiver of $A$is of Dynkin type.


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Additional Information

Pu Zhang
Affiliation: Department of Mathematics, University of Science and Technology of China, Hefei 230026, People’s Republic of China
Email: pzhang@ustc.edu.cn

DOI: https://doi.org/10.1090/S0002-9947-00-02613-1
Received by editor(s): August 3, 1998
Received by editor(s) in revised form: June 23, 1999
Published electronically: November 16, 2000
Additional Notes: Supported by the Chinese Education Ministry, the Chinese Academy of Sciences, and the National Natural Science Foundation of China.
Article copyright: © Copyright 2000 American Mathematical Society

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