coideals

Coideals are a dual notion to ideals in the context of coalgebras. Let C be a coalgebra over a field k with comultiplication Δ: C → C ⊗ C and counit ε: C → k. A linear subspace I ⊆ C is called a coideal if Δ(I) ⊆ I ⊗ C + C ⊗ I and ε(I) = 0. The condition ε(I) = 0 ensures that the quotient C/I inherits a coalgebra structure with induced maps Δ̄ and ε̄.

Equivalently, a coideal is the kernel of a surjective coalgebra morphism: if φ: C → D is a coalgebra map, then ker φ is a coideal of C, and conversely, every coideal arises as the kernel of the natural projection onto the quotient coalgebra C/I.

Examples and basic properties help illustrate the concept. The trivial subspaces {0} and C are coideals. In a simple example, if C is the polynomial-like coalgebra with Δ(x) = x ⊗ 1 + 1 ⊗ x and ε(x) = 0, then the one-dimensional subspace Ix = kx is a coideal since Δ(x) ∈ Ix ⊗ C + C ⊗ Ix and ε(x) = 0. If I and J are coideals, both their sum I+J and their intersection I ∩ J are coideals, so coideals form a lattice under inclusion.

Relation to other structures: in bialgebras and Hopf algebras, coideals that are also ideals and compatible with the antipode give rise to quotient Hopf algebras. Quotients by coideals inherit coalgebra structures, and, when stable under the additional Hopf structure, yield new Hopf algebras.

See also: coalgebra, ideal, quotient coalgebra, Hopf algebra, Hopf ideal.