Monomorphisms of Coalgebras

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, $\varphi :C\to D$ 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.

6 pages. to appear in Colloquium Mathematicum

References in corpus (1)

• On epimorphisms and monomorphisms of Hopf algebras

Cited by in corpus (2)

• On epimorphisms and monomorphisms of Hopf algebras
• A note on split extensions of bialgebras