LOCALLY DYNKIN QUIVERS AND HEREDITARY COALGEBRAS WHOSE LEFT COMODULES ARE DIRECT SUMS OF FINITE DIMENSIONAL COMODULES

ABSTRACT

Let C be an indecomposable hereditary K-coalgebra, where K is an algebraically closed field. We prove that every left C-comodule is a direct sum of finite dimensional C-comodules if and only if C is comodule Morita equivalent (see [19]) with a path K-coalgebra , where Q is a pure semisimple locally Dynkin quiver, that is, Q is either a finite quiver whose underlying graph is any of the Dynkin diagrams , , , , , , , or Q is any of the infinite quivers , , , with , shown in Sec. 2. In particular, we get in Corollaries 2.5 and 2.6 a K-coalgebra analogue of Gabriel's theorem [11] characterising representation-finite hereditary K-algebras (see also [[6], Sec. VIII.5]).

It is shown in Sec. 3 that if , then the Auslander-Reiten quiver of the category of finite dimensional left comodules has at most four connected components, and is connected if and only if Q has no sink vertices and .

ACKNOWLEDGMENT

Partially supported by Polish KBN Grant 2 P0 3A 012 16.