subcoalgebras and coideals

Let (C,Δ,ε) be a coalgebra over a field k.

Definition. Vector subspace DC is called subcoalgebra iff Δ(D)DD.

Definition. Vector subspace IC is is called coideal iff Δ(I)IC+CI and ε(I)=0.

One can show that if DC is a subcoalgebra, then (D,Δ|D,ε|D) is also a coalgebra. On the other hand, if IC is a coideal, then we can cannoicaly introduce a coalgebra structureMathworldPlanetmath on the quotient spaceMathworldPlanetmath C/I. More precisely, if xC and Δ(x)=aibi, then we define


and ε:C/Ik as ε(x+I)=ε(x). One can show that these two maps are well defined and (C/I,Δ,ε) is a coalgebra.

Title subcoalgebras and coideals
Canonical name SubcoalgebrasAndCoideals
Date of creation 2013-03-22 18:49:19
Last modified on 2013-03-22 18:49:19
Owner joking (16130)
Last modified by joking (16130)
Numerical id 4
Author joking (16130)
Entry type Definition
Classification msc 16W30