  By introducing Frobenius morphisms $F$ on algebras $A$ and their modules over
the algebraic closure ${{\bar \BF}}_q$ of the finite field $\BF_q$ of $q$
elements, we establish a relation between the representation theory of $A$ over
${{\bar \BF}}_q$ and that of the $F$-fixed point algebra $A^F$ over $\BF_q$.
More precisely, we prove that the category $\modh A^F$ 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 $\BF_q$ can be interpreted as $F$-stable
representations of a corresponding quiver over ${{\bar \BF}}_q$. We further
prove that every finite dimensional hereditary algebra over $\BF_q$ is Morita
equivalent to some $A^F$, where $A$ is the path algebra of a quiver $Q$ over
${{\bar \BF}}_q$ 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 with a given dimension
vector and to establish part of Kac's theorem for all finite dimensional
hereditary algebras over a finite field.

Rings and Algebras;Representation Theory