coideal

A coideal is a subspace of a coalgebra that is compatible with the coalgebra’s structure in a dual sense to ideals in algebras. Let C be a coalgebra over a field k with comultiplication Δ: C → C ⊗ C and counit ε: C → k. A linear subspace J ⊆ C is called a two-sided coideal if Δ(J) ⊆ J ⊗ C + C ⊗ J and ε(J) = 0. If Δ(J) ⊆ J ⊗ C, J is a left coideal; if Δ(J) ⊆ C ⊗ J, J is a right coideal. The two-sided notion is the standard one in many references.

Key property: If J is a coideal, the quotient vector space C/J inherits a natural coalgebra structure, with the induced comultiplication and counit, making the projection C → C/J a coalgebra epimorphism. Conversely, the kernel of a coalgebra morphism is a coideal. This makes coideals central to the study of quotient coalgebras and comodules.

Relation to other concepts: Coideals are dual to ideals in the sense of linear duality between algebras and coalgebras. In the setting of Hopf algebras, a subalgebra A that is also a left coideal (Δ(A) ⊆ H ⊗ A) or a right coideal (Δ(A) ⊆ A ⊗ H) is called a coideal subalgebra, a structure used in the study of quantum groups and module categories.

Examples and scope: The zero subspace is always a coideal, and many coalgebra quotients arise from coideals. Coideals provide a natural mechanism to form quotient comodules and to analyze invariant or quotient-like constructions in coalgebra and Hopf-algebra contexts.