Infinite dimensional serial algebras and their representations

We study arbitrary algebras all of whose indecomposable finite dimensional representations are uniserial, or more generally, comodules over serial coalgebras. We show that the infinite representation theory for such algebras parallels and non-trivially extends the theory of primary infinite abelian groups: we obtain a Kulikov and a Prüfer decomposability criterion and determine indecomposables in the category of locally finite modules over such an algebra. As applications, we obtain consequences on the infinite representation theory of the extended affine Dynkin quiver An˜, and the infinite line and half line quivers, and obtain Jordan form results for operators on infinite dimensional spaces. We also find a class of counterexamples to an infinite version of the pure semisimplicity conjecture: there exist algebras all of whose locally finite left modules are direct sums of indecomposables, but not all of whose right locally finite modules decompose as such. On the way, we also provide classifications of one-sided serial (co)algebras, and introduce and classify a class of quantum groups whose indecomposable representations are uniserial, namely, coserial pointed Hopf algebras. These are one-parameter deformations of group algebras and are obtained as certain Ore extensions of these.