Frobenius morphisms and representations of algebras
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By introducing Frobenius morphisms F on algebras A and their modules over the algebraic closure F q of the finite field F q of q elements, we establish a relation between the representation theory of A over F q and that of the F-fixed point algebra A F over F q . More precisely, we prove that the category 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 F q can be interpreted as F-stable representations of the corresponding quiver over F q . We further prove that every finite-dimensional hereditary algebra over F q is Morita equivalent to some A F , where A is the path algebra of a quiver Q over 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 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.
Subfields: Algebra and Number Theory, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/b76f740f8c635bf780fb642c0a92d2e5f157d75a