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comodule coalgebra
Let $H$ be a bialgebra. A right $H$ -comodule coalgebra is a coalgebra $A$ which is a right $H$ -comodule satisfying \begin{equation} (\Delta\otimes\id) t(a) = \sum a_{(1)(1)} \otimes a_{(2)(1)} \otimes a_{(1)(2)}a_{(2)(2)}, \quad (\varepsilon\otimes\id) t(a) = \varepsilon(a) \identity_H, \end{equation}for all $h \in H$ and $a \in A$ .
There is a dual notion of a $H$ -module algebra.
Example 1
Let $H$ be a Hopf algebra. Then $H$ is itself a $H$ -comodule coalgebra for the adjoint coaction $t(h) = h_{(2)} \otimes S(h_{(1)}) h_{(3)}$ .
Let $H$ be a Hopf algebra. Then $H$ is itself a $H$ -comodule coalgebra for the adjoint coaction $t(h) = h_{(2)} \otimes S(h_{(1)}) h_{(3)}$ .
comodule coalgebra is owned by mhale.
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