fundamental theorem of coalgebras

Fundamental Theorem of Coalgebras. Let (C,Δ,ε) be a coalgebra over a field k and xC. Then there exists subcoalgebra DC such that xD and dimkD<.

Proof. Let


Consider the element


Note that we may assume that (aj) are linearly independentMathworldPlanetmath and so are (ci). Let D be a subspacePlanetmathPlanetmath spanned by (bij). Of course dimkD<. Furthermore xD, because


We will show that D is a subcoalgebra, i.e. Δ(D)DD. Indeed, note that


and since ci are linearly independent we obtain that


for all i. Thus


and since aj are linearly independent, we obtain that Δ(bij)CD for all i,j. Analogously we show that Δ(bij)DC, thus


(please, see this entry ( for last equality) which completesPlanetmathPlanetmathPlanetmath the proof.

Remark. The category of finite dimensional coalgebras is dual to the category of finite dimensional algebrasMathworldPlanetmathPlanetmath (via dual spacePlanetmathPlanetmath functor), so one could think that generally they are similarPlanetmathPlanetmath. Unfortunetly Fundamental Theorem of Coalgebras is major diffrence between algebras and coalgebras. For example consider a field k and its polynomial algebra k[X]. Then whenever fk[X] is such that deg(f)>0, then a subalgebra generated by f is always infinite dimensional (if deg(f)=0 then subalgebra generated by f is k). This can never occur in coalgebras.

Title fundamental theorem of coalgebras
Canonical name FundamentalTheoremOfCoalgebras
Date of creation 2013-03-22 18:49:22
Last modified on 2013-03-22 18:49:22
Owner joking (16130)
Last modified by joking (16130)
Numerical id 6
Author joking (16130)
Entry type Theorem
Classification msc 16W30