subcoalgebra

A subcoalgebra of a coalgebra C over a field k is a subspace D ⊆ C that inherits a coalgebra structure from C via restriction. Concretely, if (Δ: C → C ⊗ C, ε: C → k) is the comultiplication and counit, then D is a subcoalgebra if Δ(D) ⊆ D ⊗ D and ε(D) ⊆ k. Equivalently, the inclusion i: D → C is a coalgebra homomorphism, and the restricted maps Δ|_D and ε|_D provide D with a coalgebra structure.

Examples include the zero subspace {0} and the whole coalgebra C. If g is a group-like element of C (Δ(g) = g ⊗ g, ε(g) = 1), then the subspace spanned by a set of group-like elements is a subcoalgebra. In finite-dimensional coalgebras, a subspace D is a subcoalgebra precisely when it is closed under Δ and ε.

Generation and construction: Given a subset S ⊆ C, the smallest subcoalgebra containing S can be formed by iteratively applying Δ and ε to S and taking spans; this yields the subcoalgebra generated by S.

Remarks: Subcoalgebras are fundamental in the study of coalgebra decompositions, coradicals, and representations. They are dual notions to subalgebras: in the finite-dimensional setting, subcoalgebras of C correspond to quotients of the dual algebra C*.