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Can you write down a general type of Hopf algebra action? How do you justify the name "action", when it is already used for group actions? There must be a common core, if the same term is used for both. In the case of a cocommutative Hopf algebra, since those are group objects in cocommutative coalgebras, I expect to recover the definition from the actions of a group object but I've got the feeling that I'm wrong.

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    $\begingroup$ Hi. I just answered your other question: mathoverflow.net/q/468267/11540. Generally, it's not good practice to ask back-to-back questions. It's better to learn from what the answer to the first question and maybe you won't need to ask the second question at all! Like groups and cogroups, you can encode what it means to be a Hopf algebra diagrammatically. Anyway, I for one cannot tell what this question is asking. Can you clarify at all? What do you mean "write down a general type of hopf algebra actions"? Yes, a Hopf algebra $H$ can act on an $H$-algebra module $A$. $\endgroup$ Apr 3 at 7:44
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    $\begingroup$ This is well-known, see, e.g., sciencedirect.com/science/article/pii/0021869386900827, or in braided categories: arxiv.org/abs/1811.10528 $\endgroup$ Apr 3 at 7:44
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    $\begingroup$ Action of a Hopf algebra on what? Since you are asking about group actions, I am thinking that you actually mean to ask about coactions instead of actions. $\endgroup$
    – J. De Ro
    Apr 3 at 7:53
  • $\begingroup$ I understand that one can define variuos things arbitrarily and ask questions about it... however the only type of action worth studying for me are group actions (maybe also monoid actions). What's the core of all the types of actions? $\endgroup$
    – user525442
    Apr 3 at 7:57
  • $\begingroup$ J. De Ro that's exactly why I've asked the questions David White mentions. I'm trying to reduce of hopf algebras to groups to make sense of them... $\endgroup$
    – user525442
    Apr 3 at 8:01

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It's not a good idea to try to "reduce" a Hopf algebra action to a group action (as the OP suggested in this comment), because a Hopf algebra is a lot more information. Instead, it's better to expand what you think about when you read the word "action" (this also answers the OP's question of "how do you justify the name "action", when it is already used for group actions").

In the category of pointed sets, a group action is a map $G\times X \to X$ making a particular diagram commute (that says $g\cdot (h\cdot x) = (gh)\cdot x$).

If $R$ is a (unital) ring, then an $R$-action is a map $R\otimes M \to M$ making $M$ into an $R$-module (this is a "left action" and of course there are also "right actions").

You can also have an action of a groupoid on a set, or of a monoid $T$ on a $T$-module.

In fact, actions can be encoded much more generally. For example, if $O$ is an operad in a symmetric monoidal category $\mathcal{C}$, then we study the category of $O$-algebras, meaning an object $X$ of $\mathcal{C}$ together with maps $O(n)\otimes_{\Sigma_n} X^{\otimes n} \to X$. There's also a notion of the action of a non-symmetric operad on an object $X$.

You can also have one category "acting" on another category, giving rise to the notion of a $\mathcal{C}$-module, for symmetric monoidal categories $\mathcal{C}$.

We have seen in the other question that coactions of cogroups can also be defined in terms of diagrams. More generally, there are comodules over coalgebras, and there are coalgebras over co-operads and comonads. One of my papers was about this.

Now, a Hopf algebra has both a multiplication $\mu: H\otimes H \to H$ and a comultiplication $\Delta: H\to H\otimes H$, making all the relevant diagrams commute. This structure assembles into something known as a bialgebra. A module $M$ over a Hopf algebra $A$ requires both an action map $a: A\otimes_k M\to M$ and a coaction map $M\to A\otimes_k M$. Here $-\otimes_k -$ is the monoidal product, and everything in sight is a $k$-module (and $A$ is a $k$-bialgebra). Furthermore:

  1. The action must be compatible with the Hopf algebra structure on $A$ just like the group action spelled out above.
  2. The coaction must be compatible with the Hopf algebra structure on $A$.
  3. The action and coaction must be compatible with each other, as spelled out in Emily's great answer here.

To come up with diagrams like this, start off by writing out equations at the level of elements, then translate those equations into diagrams, so that an equality like $g\cdot (h\cdot x) = (gh)\cdot x$ is encoded by the commutativity of the commutative diagram.

This, too, has an operadic generalization of a $P-Q$-bialgebra where $P$ and $Q$ are operads, so it's relatively easy to find the requisite commuting diagrams online, because this is a major research area. Rather than trying to reduce it to group actions, expand your mental framework!

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  • $\begingroup$ In the case of $P-Q$-bialgebras, you can compute those as $P$-algebras in $Q$-algebras, in two steps. Similarly, if you have a Hopf algebra $A$, you can let $C$ be the set of objects that have an $A$-coaction, and then look at $A$-algebras in $C$, again breaking the problem into two steps. But still it will look more like a ring action than a group action, and also acting on all the structure of a $A$-coalgebra will force that compatibility condition from Emily's answer I linked to. $\endgroup$ Apr 3 at 18:34

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