Monomorphisms of Coalgebras

TL;DR

This paper establishes new criteria for when a coalgebra morphism is a monomorphism, linking it to cohomology groups and specific algebraic conditions, with implications for Hopf algebra maps.

Contribution

It introduces novel necessary and sufficient conditions for coalgebra monomorphisms, extending existing literature and applying to Hopf algebra maps.

Findings

Coincidence of first cohomology groups characterizes monomorphisms.

A specific algebraic condition involving counit and tensor products is equivalent to monomorphism.

Provides criteria for monomorphisms in Hopf algebra maps.

Abstract

We prove new necessary and sufficient conditions for a morphism of coalgebras to be a monomorphism, different from the ones already available in the literature. More precisely, $\phi: C \to D$ is a monomorphism of coalgebras if and only if the first cohomology groups of the coalgebras $C$ and $D$ coincide if and only if $\sum_{i \in I}\epsilon(a^{i})b^{i} = \sum_{i \in I} a^{i} \epsilon(b^{i})$, for all $\sum_{i \in I}a^{i} \otimes b^{i} \in C \square_{D} C$. In particular, necessary and sufficient conditions for a Hopf algebra map to be a monomorphism are given.