Regular Representation as a Comodule
Let \(R\) be a commutative ring and \(A\) a Hopf \(R\)-algebra with comultiplication \(\Delta : A \to A \otimes_R A\) and counit \(\varepsilon : A \to R\). The pair \((A, \Delta)\) is an $A$-comodule: the coaction is \(\Delta\) itself.
Coassociativity follows from the coassociativity axiom of \(\Delta\); the counit axiom follows from the counit axiom of \(A\) as a coalgebra.
This is called the regular representation (or regular comodule) of \(A\). Geometrically, if \(A = \mathcal{O}(G)\) is the coordinate Hopf algebra of an affine group scheme \(G\), the regular representation corresponds to the action of \(G\) on its own coordinate ring by right translation. The regular representation is generally infinite-dimensional; finding finite-dimensional sub-comodules inside it is the subject of the finite generation lemma.