nLab coregular action

Let $k$ be a field, $H$ a $k$-bialgebra and $H^*$ a dual bialgebra with duality $\langle, \rangle : H\otimes H^*\to k$. The left coregular action is a Hopf action of $H$ on $H^*$ given by

$$(h,\phi) \mapsto R_h(\phi) := \phi_{(1)} \langle h, \phi_{(2)}\rangle$$

The corresponding representation $R_h: H\to End H^*$ is called the left coregular representation. It is used in the definition of Heisenberg double.