# Recovering Hopf Algebra from Group-Like Elements

Given the natural coalgebra structure on a group algebra $kG$, one can recover the group by taking the set of group-like elements of the coalgebra $kG$.

When can you go the other way? In particular, given a Hopf algebra $H$, under what conditions can one recover the structure of $H$ from it's group of group-like elements?

I'm also curious as to how the answer differs if $H$ is finitely generated versus finite dimensional.

Thanks!

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 relevant – Alexander Gruber Jan 22 at 8:05 Could you elaborate a bit on your question? If you are just given the group of group-like elements, then there is of course the group algebra over any field which is a Hopf algebra having this group of group-likes. Are you also given $H$ with its algebra structure, or do you want to know if there are other Hopf algebras with this set of group-likes, etc.? – Julian Kuelshammer Feb 7 at 13:48