Let be a -coalgebra and its right corepresentation. Recall that a sub--module is -invariant? if ; if is flat? over then is a -subcorepresentation and a -subcomodule of . If and then a -subcomodule is the same as a subcoalgebra.
A -coalgebra is cosimple if it has no subcoalgebras except for and (with ); in other words, it is the only simple object in . 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 is a field, for every -coalgebra and every -comodule , every element is contained in some finite-dimensional -subcomodule, and in particular every simple comodule is finite-dimensional and every cosimple coalgebra is finite-dimensional. A -coalgebra is cosemisimple if it is a direct sum of simple -subcoalgebras. This is equivalent to saying that every -comodule is a direct sum of simple subcomodules. A common criterion of cosemisimplicity is the existence of (say) left integrals on (left-invariant normalized functionals on ).
For example (over a field ) any group algebra is cosemisimple as a coalgebra, while the universal enveloping algebra of any nontrivial Lie -algebra is not cosemisimple. The function algebra of an affine algebraic -group is cosemisimple iff is linearly reductive; over a transcendental parameter of deformation, this is preserved for quantized function algebras (cf. quantum group).