Frobenius morphisms and representations of algebras

Abstract

By introducing Frobenius morphisms F F on algebras A A and their modules over the algebraic closure F ¯ q {\overline {\mathbb {F}}}_q of the finite field F q {\mathbb {F}}_q of q q elements, we establish a relation between the representation theory of A A over F ¯ q \overline {\mathbb {F}}_q and that of the F F -fixed point algebra A F A^F over F q {\mathbb {F}}_q . More precisely, we prove that the category \textbf {mod}- A F \text {\textbf {mod}-}A^F of finite-dimensional A F A^F -modules is equivalent to the subcategory of finite-dimensional F F -stable A A -modules, and, when A A is finite dimensional, we establish a bijection between the isoclasses of indecomposable A F A^F -modules and the F F -orbits of the isoclasses of indecomposable A A -modules. Applying the theory to representations of quivers with automorphisms, we show that representations of a modulated quiver (or a species) over F q {\mathbb {F}}_q can be interpreted as F F -stable representations of the corresponding quiver over F ¯ q \overline {\mathbb {F}}_q . We further prove that every finite-dimensional hereditary algebra over F q {\mathbb {F}}_q is Morita equivalent to some A F A^F , where A A is the path algebra of a quiver Q Q over F ¯ q \overline {\mathbb {F}}_q and F F is induced from a certain automorphism of Q Q . A close relation between the Auslander-Reiten theories for A A and A F A^F is established. In particular, we prove that the Auslander-Reiten (modulated) quiver of A F A^F is obtained by “folding" the Auslander-Reiten quiver of A A . Finally, by taking Frobenius fixed points, we are able to count the number of indecomposable representations of a modulated quiver over F q {\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.