What I learned today: Coadjoint action of group Hopf algebras

This is another post in my ``psuedo-daily'' series ``What I learned Today''

Yesterday we talked about actions of a Hopf algebra $H$ on an algebra $A$. Today, let's talk about some examples of this (I didn't cover as much ground as I hoped to today, but we're trying to make these blogs daily!). Every Hopf algebra acts on itself by

$\displaystyle h \triangleright g = \sum_h h_{(1)} g Sh_{(2)}$

This is the adjoint action. It's not hard to prove that it's 1) an action and 2) a Hopf action (that it plays well with the Hopf algebra structure of $H$ and the algebra structure of $A$). As a more concrete example, let's look at the group Hopf algebra $H=kG$ for a finite group $G$. The coproduct is given by $\Delta g = g \otimes g$ and the antipode is group inversion: $Sg = g^-1$. In this case, the adjoint action becomes:

$\displaystyle h \triangleright g = hgh^{-1}$

Which is just group conjugation.

Hopf algebras can act just as well on coalgebras - in this case we require that the action commutes with the coproduct of $A$. Now if $H'$ and $H$ are dually paired, then we also have adjoint action, or rather, a coadjoint action of $H'$ on the coalgebra $H$, given by:

$\displaystyle \phi \triangleright h = \sum_h h_{(2)} \langle \phi, (Sh_{(1)})h_{(3)} \rangle$

We know that the algebra of functions on $G$, $k(G)$ is the Hopf algebra dual of $kG$, where $\langle f, g \rangle = f(g)$, so let's see what the this action becomes for $H' = k(G)$ and $H = kG$:

$\displaystyle f \triangleright g = g \langle f, (Sg)g \rangle = f(1) \cdot g$

Additionally, we also have the notion of a coaction. Generally, if $H$ is a Hopf algebra and $A$ an algebra, we say $\beta : A \rightarrow H \otimes A$ is a coaction of $H$ on $A$ (or that $A$ is a $H$-comodule algebra if $\beta$ is a algebra morphism, $(I \otimes \beta) \circ \beta = (\Delta \otimes I) \circ \beta$ and $I = (\epsilon \otimes I) \circ \beta$, where $I$ is the identity and $\Delta$ and $\epsilon$ are the coalgebra structure maps.

The most interesting case of this is the quantum group $SL_q(2)$ coacting on the quantum plane $\mathbb{A}^2_q$. This is a topic for another post, though.