Frobenius morphisms and representations of algebras

Author:

Deng Bangming,Du Jie

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.

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

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