[math] \newcommand{\mathds}{\mathbb}[/math]

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Given a finite dimensional Hopf algebra [math]A[/math], prove that its dual [math]A^*[/math] is a Hopf algebra too, with structural maps as follows:

[[math]] \Delta^t:A^*\otimes A^*\to A^* [[/math]]

[[math]] \varepsilon^t:\mathbb C\to A^* [[/math]]

[[math]] m^t:A^*\to A^*\otimes A^* [[/math]]

[[math]] u^t:A^*\to\mathbb C [[/math]]

[[math]] S^t:A^*\to A^* [[/math]]

Also, check that [math]A[/math] is commutative if and only if [math]A^*[/math] is cocommutative, and also discuss what happens in the cases [math]A=C(G)[/math] and [math]A=C^*(H)[/math], with [math]G,H[/math] being finite groups.