fundamental theorem of coalgebras
Fundamental Theorem of Coalgebras. Let be a coalgebra over a field and . Then there exists subcoalgebra such that and .
Consider the element
We will show that is a subcoalgebra, i.e. . Indeed, note that
and since are linearly independent we obtain that
for all . Thus
and since are linearly independent, we obtain that for all . Analogously we show that , thus
(please, see this entry (http://planetmath.org/TensorProductOfSubspacesOfVectorSpaces) for last equality) which completes the proof.
Remark. The category of finite dimensional coalgebras is dual to the category of finite dimensional algebras (via dual space functor), so one could think that generally they are similar. Unfortunetly Fundamental Theorem of Coalgebras is major diffrence between algebras and coalgebras. For example consider a field and its polynomial algebra . Then whenever is such that , then a subalgebra generated by is always infinite dimensional (if then subalgebra generated by is ). This can never occur in coalgebras.
|Title||fundamental theorem of coalgebras|
|Date of creation||2013-03-22 18:49:22|
|Last modified on||2013-03-22 18:49:22|
|Last modified by||joking (16130)|