Frobenius morphisms and representations of algebras

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 $\text {\textbf {mod}-}A^F$ 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.