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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Representations as elements in affine composition algebras
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by Pu Zhang PDF
Trans. Amer. Math. Soc. 353 (2001), 1221-1249 Request permission


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 $|k| \ne 2, 3$, 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
  • MR Author ID: 260913
  • Email:
  • 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.
  • © Copyright 2000 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 353 (2001), 1221-1249
  • MSC (2000): Primary 16G20, 17B37; Secondary 16G60, 17B40, 05A30, 18G15
  • DOI:
  • MathSciNet review: 1707704