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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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  • $\begingroup$ relevant $\endgroup$ – Alexander Gruber Jan 22 '13 at 8:05
  • $\begingroup$ 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.? $\endgroup$ – Julian Kuelshammer Feb 7 '13 at 13:48
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Regarding the question:

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?

the simplest case in which this happens corresponds to f.d. cocommutative hopf algebras. In particular, it can be shown that:

If $H$ is a finite dimensional, cocommutative hopf algebra, over an algebraically closed field of characteristic zero, then: $$ H\cong kG(H) $$ where $G(H)$ is the set of the grouplikes of $H$, which is a group, and $kG(H)$ is its group algebra.

This is actually a consequence of the Cartier-Konstant-Milnor-Moore theorem (when applied to the finite dimensional case).

Another interesting situation arises in the case of the commutative hopf algebras:

If $H$ is a finite dimensional, commutative hopf algebra, over an algebraically closed field of characteristic zero, then: $$ H\cong \big(kG(H^*)\big)^* $$ where $H^*$ is the dual hopf algebra of $H$, $G(H^*)$ is the set of the grouplikes of $H^*$, $kG(H^*)$ its group hopf algebra and $\big(kG(H^*)\big)^*$ the dual hopf algebra of the group hopf algebra $kG(H^*)$.

A special case combining both commutativity and cocommutativity is discussed in Reference on correspondence between commutative Hopf Algebras and Groups. Maybe that could also be of some interest for the OP.

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    $\begingroup$ So the second statement could also be formulated as "any finite group scheme over an algebraically closed field of characteristic $0$ is just a finite group". This also indicates why it fails to hold on positive characteristic. $\endgroup$ – Tobias Kildetoft May 15 '18 at 6:51
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In general you can't. In the case of pointed hopf algebras (certainly a big glass of hopf algebras) where the group likes are the coradical, the strucure is basically dictated by the group likes and by a Nichols algebra (the bosonization of both actually gives the graded algebra associated to the coradical filtration of your hopf algebra). Both ingredients are independent, you can actually use the same nichols algebra for different groups, and then have different hopf algebras with the same group-likes. If you're interested, google "the lifting method for hopf algebras" and you'll get a lot of detailed information about what I outlined here.

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  • $\begingroup$ But, if i correctly understand your point, the case you are describing corresponds to the "opposite" situation of the one described in the OP: that is the case for which the hopf algebra cannot be recovered from its grouplikes. $\endgroup$ – KonKan May 13 '18 at 21:47

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