# 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

$\left(h,\varphi \right)↦{R}_{h}\left(\varphi \right):={\varphi }_{\left(1\right)}⟨h,{\varphi }_{\left(2\right)}⟩$(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.