Coadjoint representation of a Hopf algebra

by Lullaby   Last Updated August 03, 2019 11:20 AM

I have problems with understanding the concept of the coadjoint action of a Hopf algebra.

Let me recall something easier. The (left) adjoint action of a Hopf algebra on itself is defined as follows: it is a Hopf algebra homomorphism $ad: H \rightarrow End(H)$, where $ad_{a}(b)=\sum \limits_{(a)}a_{(1)}bS(a_{(2)})$ for any $a,b \in H$ and $S$ is the antipode of $H$.

Similarly we can define the right adjoint action, which is given by $ad_{a}(b)=\sum \limits_{(a)}S(a_{(1)})ba_{(2)}$ for $a,b \in H$.

Now, in order to construct the dual representation of the adjoint representation, we need to take the dual of the Hopf algebra H, namely $H^{*}=Hom_{\mathbb{K}}(H, \mathbb{K})$. Then the left coadjoint action of $H$ is the Hopf algebra homomorphism $coad: H \rightarrow End(H^{*})$.

Can anybody help me with understanding this definition? Thank you very much in advance.

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