Convolution of Algebra Homomorphisms

Let \(R\) be a commutative ring, \(A\) an \(R\)-bialgebra (in particular, an \(R\)-coalgebra with comultiplication \(\Delta : A \to A \otimes_R A\)), and \(B\) a commutative \(R\)-algebra with multiplication \(m_B : B \otimes_R B \to B\). For two \(R\)-algebra homomorphisms \(f, g : A \to B\), their convolution \(f * g : A \to B\) is the \(R\)-linear map

\[ f * g \;=\; m_B \circ (f \otimes_R g) \circ \Delta. \]

Using sumless Sweedler notation \(\Delta(a) = \sum_{(a)} a_{(1)} \otimes a_{(2)}\), the formula reads

\[ (f * g)(a) \;=\; \sum_{(a)} f(a_{(1)}) \cdot g(a_{(2)}). \]

Convolution restricts the general \(\operatorname{Hom}_R(A, B)\)-level convolution from [[node:foundations.hopf_algebras|the Hopf algebra infrastructure]] (Mathlib's \(\texttt{Mathlib.RingTheory.Coalgebra.Convolution}\)) to the subset of \(R\)-algebra homomorphisms. That this subset is closed under convolution is the content of the next theorem.