convolution product

Suppose you have a coalgebra C (you can split elements) and an algebra A (you can multiply elements). Then any two linear maps from C to A can be combined: split the input with the coproduct, send each piece through one of the maps, and multiply the results back together in A. This recipe is the convolution product, and it makes the whole space of linear maps Hom(C, A) into an algebra in its own right.

Formally, for f, g ∈ Hom(C, A) the convolution f * g is defined by f * g = m_A ∘ (f ⊗ g) ∘ Δ_C. In Sweedler notation, (f * g)(c) = Σ f(c(1)) g(c(2)). With this product Hom(C, A) is an associative algebra; its unit is the composite u_A ∘ ε_C, the map sending c to ε(c) 1_A. Associativity of convolution follows from coassociativity of Δ and associativity of m.

Convolution is the conceptual home of the antipode: in a bialgebra H, the antipode is by definition the convolution inverse of the identity map in Hom(H, H). The name comes from classical analysis — the convolution of functions on a group, (f * g)(x) = Σ f(y) g(y^{-1}x), is exactly this construction for the Hopf algebra of functions on a finite group, where Δ encodes the group law.