nLab cosemisimple coalgebra

Let $C$ be a $k$-coalgebra and $\rho:V\to V\otimes C$ its right corepresentation. Recall that a sub-$k$-module $W\subset V$ is $\rho$-invariant if $Im \rho|_W\subset W\otimes C$; if $C$ is flat? over $k$ then $\rho|_W$ is a $C$-subcorepresentation and $(W,\rho|_W)$ a $C$-subcomodule of $(V,\rho_V)$. If $V=C$ and $\rho=\Delta_C$ then a $C$-subcomodule is the same as a subcoalgebra.

A $k$-coalgebra $C$ is cosimple if it has no subcoalgebras except for $C$ and $0$ (with $C \neq 0$); in other words, it is the only simple object in $Comod^C$. Emphasising ‘cosimple’ instead of ‘simple’ is convenient because, for Hopf algebras, both semisimplicity and cosemisimplicity make sense. It is a basic fact (not paralleled in module theory) that if $k$ is a field, for every $k$-coalgebra $C$ and every $C$-comodule $(V,\rho)$, every element $v\in V$ is contained in some finite-dimensional $C$-subcomodule, and in particular every simple comodule is finite-dimensional and every cosimple coalgebra is finite-dimensional. A $k$-coalgebra $C$ is cosemisimple if it is a direct sum of simple $k$-subcoalgebras. This is equivalent to saying that every $C$-comodule is a direct sum of simple subcomodules. A common criterion of cosemisimplicity is the existence of (say) left integrals on $H$ (left-invariant normalized functionals on $H$).

For example (over a field $k$) any group algebra $k[G]$ is cosemisimple as a coalgebra, while the universal enveloping algebra $U(g)$ of any nontrivial Lie $k$-algebra $g\neq 0$ is not cosemisimple. The function algebra $\mathcal{O}(G)$ of an affine algebraic $k$-group is cosemisimple iff $G$ is linearly reductive; over a transcendental parameter $q$ of deformation, this is preserved for quantized function algebras (cf. quantum group).