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 $\mathrm{Im}\rho {\mid }_W\subset W\otimes C$; if $C$ is flat? over $k$ then $\rho {\mid }_W$ is a $C$-subcorepresentation and $(W,\rho {\mid }_W)$ 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 $C\ne 0$); in other words, it is the only simple object in ${\mathrm{Comod}}^C$. 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 $g\ne 0$ is not cosemisimple. The function algebra $𝒪(G)$ 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).