Monomorphisms of coalgebras

A. L. Agore ¹

¹ Faculty of Mathematics and Computer Science University of Bucharest Str. Academiei 14 RO-010014 Bucureşti 1, Romania and Department of Mathematics Academy of Economic Studies Piata Romana 6 RO-010374 Bucureşti 1, Romania

Colloquium Mathematicum, Tome 120 (2010) no. 1, pp. 149-155

Résumé

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.

Détail

DOI : 10.4064/cm120-1-11

Mots-clés : prove necessary sufficient conditions morphism coalgebras monomorphism different already available literature precisely varphi rightarrow monomorphism coalgebras only first cohomology groups coalgebras coincide only sum varepsilon sum varepsilon sum otimes mathbin square particular necessary sufficient conditions hopf algebra map monomorphism given