Coxeter transformation and inverses of Cartan matrices for coalgebras

TL;DR

This paper investigates Coxeter transformations and their duals in the context of pointed Euler coalgebras, relating these transformations to almost split sequences and dimension vectors of comodules, extending classical representation theory concepts.

Contribution

It introduces a detailed study of Coxeter transformations for coalgebras and establishes their connection with almost split sequences and dimension vectors in this setting.

Findings

Existence of unique almost split sequences ending at certain comodules.

Relation between Coxeter transformations and dimension vectors in hereditary coalgebras.

Extension of classical representation theory results to coalgebra context.

Abstract

Let C be a coalgebra and consider the Grothendieck groups of the categories of the socle-finite injective right and left C-comodules. One of the main aims of the paper is to study Coxeter transformation, and its dual, of a pointed sharp Euler coalgebra C, and to relate the action of these transformations on a class of indecomposable finitely cogenerated C-comodules N with almost split sequences starting or ending with N. We also show that if C is a pointed K-coalgebra such that the every vertex of the left Gabriel quiver of C has only finitely many neighbours, then for any indecomposable non-projective left C-comodule N of finite K-dimension, there exists a unique almost split sequence of finitely cogenerated left C-comodules ending at N. We show that the dimension vector of the Auslander-Reiten translate given by the Coxeter transformation, if C is hereditary, or more generally, if inj.dim DN=1 and Hom(C,DN)=0.