bialgebra

A bialgebra is a vector space wearing two hats at once: it is an algebra (you can multiply elements) and a coalgebra (you can comultiply them), and crucially the two structures are compatible. Compatibility means the comultiplication is itself an algebra map — splitting a product is the same as multiplying the splittings. This single coherence condition fuses the two worlds into one.

Formally, over k, a bialgebra is a tuple (H, m, u, Δ, ε) where (H, m, u) is an associative unital algebra and (H, Δ, ε) is a coalgebra, such that Δ : H -> H ⊗ H and ε : H -> k are algebra homomorphisms. Equivalently, m : H ⊗ H -> H and u : k -> H are coalgebra homomorphisms; the two phrasings are the same axiom. Here H ⊗ H carries the tensor-product algebra and coalgebra structures.

Bialgebras sit one step below Hopf algebras: a Hopf algebra is a bialgebra with an extra map, the antipode. Many natural objects are bialgebras that happen to also be Hopf algebras — group algebras, universal enveloping algebras, the polynomial ring k[x] with Δ(x) = x ⊗ 1 + 1 ⊗ x — but there are bialgebras with no antipode, for instance the polynomial bialgebra k[N] of functions on the multiplicative monoid of natural numbers.