Frobenius morphisms and representations of algebras
Abstract
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.