Frobenius morphisms and representations of algebras

Original · EN

By introducing Frobenius morphisms F on algebras A and their modules over the algebraic closure q of the finite field of q elements, we establish a relation between the representation theory of A over q and that of the F-fixed point algebra Aᶠ over. More precisely, we prove that the category Aᶠ of finite dimensional Aᶠ-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 Aᶠ-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 can be interpreted as F-stable representations of a corresponding quiver over q. We further prove that every finite dimensional hereditary algebra over is Morita equivalent to some Aᶠ, where A is the path algebra of a quiver Q over q and F is induced from a certain automorphism of Q. A close relation between the Auslander-Reiten theories for A and Aᶠ is established. In particular, we prove that the Auslander-Reiten (modulated) quiver of Aᶠ 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.