nLab
coregular action

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

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

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