nLab coideal

Given a commutative unital ring $k$ and a coassociative $k$-coalgebra $C = (C,\Delta,\epsilon)$ with comultiplication $\Delta$ and counit $\epsilon\colon C\to k$, a $k$-submodule $I\subseteq C$ is

• a right coideal if $\Delta(I)\subseteq \overline{I\otimes C}$,

• a left coideal if $\Delta(I)\subseteq \overline{C\otimes I}$,

• a coideal if $\Delta(I)\subseteq \overline{I\otimes C}+\overline{C\otimes I}$ and $\epsilon(I)=0$.

Here, $\overline{I \otimes C}$ denotes the image of $I \otimes C \to C \otimes C$ (this distinction is not necessary when $C$ is a flat $k$-module, for example when $k$ is a field).

In other words, a left coideal is simply a left subcomodule in $C$ with coaction being the comultiplication and a coideal is a subbicomodule in $C$, where the comultiplication plays simultaneously the roles of left and right $C$-coactions on itself.

If $I\subseteq C$ is a coideal, then the quotient $k$-module $C/I$ is equipped with the canonical structure of a coassociative $k$-coalgebra, the quotient co(al)gebra; the comultiplication is induced via choosing an arbitrary representative in each class, namely

$$\Delta_{C/I}(c+I):= \Delta_C(c) + I\otimes C + C\otimes I \in C/I \otimes C/I,$$

or, in Sweedler notation, $\Delta_{C/I}(c) = \sum (c_{(1)}+I)\otimes (c_{(2)}+I)$ where $\Delta_C(c)= \sum c_{(1)}\otimes c_{(2)}$.