Monomorphisms of coalgebras
Tom 120 / 2010
Colloquium Mathematicum 120 (2010), 149-155
MSC: 16T15, 16T05.
DOI: 10.4064/cm120-1-11
Streszczenie
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, $\varphi: C \rightarrow D$ is a monomorphism of coalgebras if and only if the first cohomology groups of the coalgebras $C$ and $D$ coincide if and only if $\sum_{i \in I}\varepsilon(a^{i})b^{i} = \sum_{i \in I} a^{i} \varepsilon(b^{i})$ for all $\sum_{i \in I}a^{i} \otimes b^{i} \in C \mathbin\square_{D} C$. In particular, necessary and sufficient conditions for a Hopf algebra map to be a monomorphism are given.
