Hopf algebras actions

Question

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.

Answer 1

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!