Algebra Homomorphisms Form a Group Under Convolution
Let \(R\) be a commutative ring, \(A\) an \(R\)-Hopf algebra with counit \(\varepsilon : A \to R\) and antipode \(S : A \to A\), and \(B\) a commutative \(R\)-algebra. The set \(\operatorname{Hom}_{R\text{-Alg}}(A, B)\) of \(R\)-algebra homomorphisms forms a group under convolution, with:
- identity element the composition \(\eta_B \circ \varepsilon : A \to B\), where \(\eta_B : R \to B\) is the structure map of \(B\) as an \(R\)-algebra;
- inverse of \(f : A \to B\) the composition \(f \circ S : A \to B\).
Proof
That the convolution lands in algebra homs is the preceding theorem. That \(\eta_B \circ \varepsilon\) and \(f \circ S\) are algebra homs is direct (counit and antipode are coalgebra/anti-algebra maps; combined with the commutativity of \(B\), the antipode composed with \(f\) remains an algebra hom — the antipode-is-anti-algebra-hom and commutativity-of-\(B\) cancel).
Identity. For any \(f \in \operatorname{Hom}_{R\text{-Alg}}(A, B)\) and \(a \in A\),
\[ ((\eta_B \varepsilon) * f)(a) \;=\; \sum_{(a)} \eta_B(\varepsilon(a_{(1)})) \cdot f(a_{(2)}) \;=\; \sum_{(a)} f\bigl(\varepsilon(a_{(1)})\, a_{(2)}\bigr) \;=\; f(a) \]
by the counit axiom \(\sum_{(a)} \varepsilon(a_{(1)})\, a_{(2)} = a\) of the coalgebra \(A\) (and \(f\) being \(R\)-linear). Similarly \(f * (\eta_B \varepsilon) = f\).
Inverse. For \(f \in \operatorname{Hom}_{R\text{-Alg}}(A, B)\) and \(a \in A\),
\[ (f * (f \circ S))(a) \;=\; \sum_{(a)} f(a_{(1)}) \cdot f(S(a_{(2)})) \;=\; f\bigl(\sum_{(a)} a_{(1)} S(a_{(2)})\bigr) \;=\; f(\eta_A \varepsilon(a)) \;=\; \eta_B \varepsilon(a) \]
by the antipode axiom \(\sum_{(a)} a_{(1)} S(a_{(2)}) = \eta_A \varepsilon(a)\), where \(\eta_A : R \to A\) is the unit. Hence \(f * (f \circ S) = \eta_B \varepsilon\), the identity element. Symmetrically \((f \circ S) * f = \eta_B \varepsilon\).
Associativity. Follows from the coassociativity of \(\Delta\) and the associativity of the multiplication of \(B\) by a direct computation with Sweedler notation.
Hence \((\operatorname{Hom}_{R\text{-Alg}}(A, B), *)\) is a group.
\(\square\)