# 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 $\left(W,\rho {\mid }_{W}\right)$ a $C$-subcomodule of $\left(V,{\rho }_{V}\right)$. 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 $\left(V,\rho \right)$, 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 $k\left[G\right]$ is cosemisimple as a coalgebra, while the universal enveloping algebra $U\left(g\right)$ of any nontrivial Lie $k$-algebra $g\ne 0$ is not cosemisimple. The function algebra $𝒪\left(G\right)$ 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).

Revised on August 16, 2009 19:15:32 by Toby Bartels (71.104.230.172)