# 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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