Path subcoalgebras, finiteness properties and quantum groups

TL;DR

This paper classifies certain path subcoalgebras related to quivers, studies their finiteness and co-Frobenius properties, and explores their potential to form quantum groups with non-zero integrals.

Contribution

It provides a classification of co-Frobenius path subcoalgebras and characterizes which can be endowed with Hopf algebra structures to form quantum groups.

Findings

Co-Frobenius path subcoalgebras are sums of specific quiver-based subcoalgebras.

Identifies conditions under which these coalgebras admit Hopf algebra structures.

Classifies quantum groups arising from these coalgebras over fields with roots of unity.

Abstract

We study subcoalgebras of path coalgebras that are spanned by paths (called path subcoalgebras) and subcoalgebras of incidence coalgebras, and propose a unifying approach for these classes. We discuss the left quasi-co-Frobenius and the left co-Frobenius properties for these coalgebras. We classify the left co-Frobenius path subcoalgebras, showing that they are direct sums of certain path subcoalgebras arising from the infinite line quiver or from cyclic quivers. We also discuss the coreflexive property for the considered classes of coalgebras. Finally, we investigate which of the co-Frobenius path subcoalgebras can be endowed with Hopf algebra structures, in order to produce some quantum groups with non-zero integrals, and classify all these structures over a field with primitive roots of unity of any order. These turn out to be liftings of quantum lines over certain not necessarily abelian groups.