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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


Frobenius morphisms and representations of algebras

Authors: Bangming Deng and Jie Du
Journal: Trans. Amer. Math. Soc. 358 (2006), 3591-3622
MSC (2000): Primary 16G10, 16G20, 16G70
Published electronically: January 24, 2006
MathSciNet review: 2218990
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Abstract: By introducing Frobenius morphisms $ F$ on algebras $ A$ and their modules over the algebraic closure $ {\overline {\mathbb{F}}}_q$ of the finite field $ {\mathbb{F}}_q$ of $ q$ elements, we establish a relation between the representation theory of $ A$ over $ \overline {\mathbb{F}}_q$ and that of the $ F$-fixed point algebra $ A^F$ over $ {\mathbb{F}}_q$. More precisely, we prove that the category    mod-$ A^F$ of finite-dimensional $ A^F$-modules is equivalent to the subcategory of finite-dimensional $ F$-stable $ A$-modules, and, when $ A$ is finite dimensional, we establish a bijection between the isoclasses of indecomposable $ A^F$-modules and the $ F$-orbits of the isoclasses of indecomposable $ A$-modules. Applying the theory to representations of quivers with automorphisms, we show that representations of a modulated quiver (or a species) over $ {\mathbb{F}}_q$ can be interpreted as $ F$-stable representations of the corresponding quiver over $ \overline {\mathbb{F}}_q$. We further prove that every finite-dimensional hereditary algebra over $ {\mathbb{F}}_q$ is Morita equivalent to some $ A^F$, where $ A$ is the path algebra of a quiver $ Q$ over $ \overline {\mathbb{F}}_q$ and $ F$ is induced from a certain automorphism of $ Q$. A close relation between the Auslander-Reiten theories for $ A$ and $ A^F$ is established. In particular, we prove that the Auslander-Reiten (modulated) quiver of $ A^F$ is obtained by ``folding" the Auslander-Reiten quiver of $ A$. Finally, by taking Frobenius fixed points, we are able to count the number of indecomposable representations of a modulated quiver over $ {\mathbb{F}}_q$ with a given dimension vector and to generalize Kac's theorem for all modulated quivers and their associated Kac-Moody algebras defined by symmetrizable generalized Cartan matrices.

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Additional Information

Bangming Deng
Affiliation: Department of Mathematics, Beijing Normal University, Beijing 100875, People’s Republic of China

Jie Du
Affiliation: School of Mathematics, University of New South Wales, Sydney 2052, Australia

PII: S 0002-9947(06)03812-8
Received by editor(s): August 14, 2003
Received by editor(s) in revised form: August 12, 2004
Published electronically: January 24, 2006
Additional Notes: This work was partially supported by the NSF of China (Grant no. 10271014), the Doctoral Program of Higher Education, and the Australian Research Council.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.