Coactions

\(\newcommand{\rar}{\rightarrow} \newcommand{\mb}{\mathbb} \newcommand{\mf}{\mathfrak}\)Just as we can dualize things to turn an algebra into a coalgebra, we can dualize an action to get a coaction.
A left coaction of a set \(S\) on a space \(V\) is usually written as a map \(\beta: V \rar S \otimes V\), which we compare to an action of \(T\) on \(V\) which is written \(T \otimes V \rar V\). If \(V\) is finite-dimensional, then we can turn the coaction map into a map \(End(V^\vee) \rar S\). We say that \(V\) is an \(S\)-comodule with coaction map \(\beta\).
If \(S\) can be dualized, for instance if it is a finite-dimensional vector space, then we can interpret a coaction of \(S\) as an action of \(S^\vee\). In particular, if the coaction of \(S\) is the map \(\beta: V \rar S\otimes V\) then the induced action of \(S^\vee\) is given by \(\rho(t \otimes v) = (t \otimes id)(\beta(v))\). Dually, an action of \(S^\vee\) gives a coaction of \(S\), by setting \(\beta(v) = e_i\otimes \rho(e^i \otimes v)\) where \(e_i\) is a basis of \(S\) and \(e^i\) is the corresponding dual basis of \(S^\vee\).

We can also talk about algebra coactions and coalgebra coactions. The rules for an algebra coaction are dual to the rules for a coalgebra action. Namely, if we have an algebra \(A\) and two coactions of \(A\) labeled \((V, \beta)\) and \((W, \gamma)\), we can make a algebra coaction on \(V\otimes W\) by setting
$$\beta \hat \otimes \gamma = (m \otimes id \otimes id) \circ (id \otimes \sigma_{VA}\otimes id) \circ(\beta \otimes \gamma)$$In other words, if \(\beta(v) = a_i \otimes v^i\) and \(\gamma(w) = a_j \otimes w^j\), then we get that
$$(\beta \hat \otimes \gamma)(v \otimes w) = a_ia_j \otimes v^i \otimes w^j$$
The rules for a coalgebra coaction are dual to the rules for an algebra action. In other words,
$$(\Delta \otimes id) \circ \beta = (id \otimes \beta) \circ \beta$$

Just to establish it, we say that the trivial coaction of a unital thing on \(V\) sends \(v \in V\) to \(1 \otimes v\), just as the trivial action of a counital thing was the map \(s\otimes v\mapsto \epsilon(s)v\).

Coactions are often useful when the coacting set doesn't have a nice dual, perhaps due to being infinite-dimensional. Sometimes the coacting set is simply nicer to work with. For instance, instead of dealing with a group algebra, which can be badly behaved for infinite groups like Lie groups, we can instead look at the representation ring \(Rep(\mb kG)\) of \(mb kG\), i.e. the algebra generated by the set of functions \(\pi_i^j\) where \(\pi\) is the representation map for some finite-dimensional representation and \(\pi_i^j(\hat g)\) is the \((i,j)\) entry of \(\pi(\hat g)\).
Notably, if \(\rho\) is the representation map of some faithful finite-dimensional representation, then \(Rep(\mb kG)\) is generated by the \(\rho_i^j\). We'll talk about \(Rep\) a bit more later.

And just to show again that the notion of a coaction is really dual to actions, here's a bunch of diagrams; the solid line is the coacting set, the dashed is the set being coacted upon. The coaction itself is given by a black box. If you compare with the action diagrams, you'll see that they are in fact dual.