2003/07/18 by Bangming Deng, Jie Du, Deng, Bangming +1 · 2 citations
Mathematics · #16G20 #20G05 #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA) #math.RA #math.RT #msc:16G20 #msc:20G05
paper · pdf · doi:10.48550/arxiv.math/0307256
28 pages
arxiv created 2003/07/18 · arxiv updated 2009/12/01
By introducing Frobenius morphisms F on algebras A and their modules over the algebraic closure \BFq of the finite field \BFq of q elements, we establish a relation between the representation theory of A over \BFq and that of the F-fixed point algebra AF over \BFq. More precisely, we prove that the category \modh 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 \BFq can be interpreted as F-stable representations of a corresponding quiver over \BFq. We further prove that every finite dimensional hereditary algebra over \BFq is Morita equivalent to some AF, where A is the path algebra of a quiver Q over \BFq 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.