# subcoalgebras and coideals

Let $(C,\Delta,\varepsilon)$ be a coalgebra over a field $k$.

Definition. Vector subspace $D\subseteq C$ is called subcoalgebra iff $\Delta(D)\subseteq D\otimes D$.

Definition. Vector subspace $I\subseteq C$ is is called coideal iff $\Delta(I)\subseteq I\otimes C+C\otimes I$ and $\varepsilon(I)=0$.

One can show that if $D\subseteq C$ is a subcoalgebra, then $(D,\Delta_{|D},\varepsilon_{|D})$ is also a coalgebra. On the other hand, if $I\subseteq C$ is a coideal, then we can cannoicaly introduce a coalgebra structure  on the quotient space  $C/I$. More precisely, if $x\in C$ and $\Delta(x)=\sum a_{i}\otimes b_{i}$, then we define

 $\Delta^{\prime}:C/I\to(C/I)\otimes(C/I);$
 $\Delta^{\prime}(x+I)=\sum(a_{i}+I)\otimes(b_{i}+I)$

and $\varepsilon^{\prime}:C/I\to k$ as $\varepsilon^{\prime}(x+I)=\varepsilon(x)$. One can show that these two maps are well defined and $(C/I,\Delta^{\prime},\varepsilon^{\prime})$ is a coalgebra.

Title subcoalgebras and coideals SubcoalgebrasAndCoideals 2013-03-22 18:49:19 2013-03-22 18:49:19 joking (16130) joking (16130) 4 joking (16130) Definition msc 16W30