nLab
cosemisimple coalgebra

Let CC be a kk-coalgebra and ρ:VVC\rho:V\to V\otimes C its right corepresentation. Recall that a sub-kk-module WVW\subset V is ρ\rho-invariant? if Imρ WWCIm \rho|_W\subset W\otimes C; if CC is flat? over kk then ρ W\rho|_W is a CC-subcorepresentation and (W,ρ W)(W,\rho|_W) a CC-subcomodule of (V,ρ V)(V,\rho_V). If V=CV=C and ρ=Δ C\rho=\Delta_C then a CC-subcomodule is the same as a subcoalgebra.

A kk-coalgebra CC is cosimple if it has no subcoalgebras except for CC and 00 (with C0C \neq 0); in other words, it is the only simple object in Comod CComod^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 kk is a field, for every kk-coalgebra CC and every CC-comodule (V,ρ)(V,\rho), every element vVv\in V is contained in some finite-dimensional CC-subcomodule, and in particular every simple comodule is finite-dimensional and every cosimple coalgebra is finite-dimensional. A kk-coalgebra CC is cosemisimple if it is a direct sum of simple kk-subcoalgebras. This is equivalent to saying that every CC-comodule is a direct sum of simple subcomodules. A common criterion of cosemisimplicity is the existence of (say) left integrals on HH (left-invariant normalized functionals on HH).

For example (over a field kk) any group algebra k[G]k[G] is cosemisimple as a coalgebra, while the universal enveloping algebra U(g)U(g) of any nontrivial Lie kk-algebra g0g\neq 0 is not cosemisimple. The function algebra 𝒪(G)\mathcal{O}(G) of an affine algebraic kk-group is cosemisimple iff GG is linearly reductive; over a transcendental parameter qq 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)