module algebra
When a group acts on a space, it should act by symmetries — respecting whatever structure the space carries. A module algebra is the Hopf-algebraic version of this idea: it is an algebra on which a Hopf algebra acts, and the action respects the algebra's multiplication and unit. The coproduct is precisely the device that tells you how to spread a single Hopf-algebra action across a product of two elements.
Formally, let H be a Hopf algebra. An H-module algebra is an algebra A that is also an H-module such that h · (ab) = Σ (h(1) · a)(h(2) · b) and h · 1_A = ε(h) 1_A for all h ∈ H, a, b ∈ A. The first axiom is a generalized Leibniz rule: a grouplike element acts as an algebra automorphism, while a primitive element acts as a derivation, since then h · (ab) = (h · a) b + a (h · b).
This unifies two familiar notions. Group actions by automorphisms are exactly k[G]-module algebra structures, and actions of a Lie algebra by derivations are exactly U(g)-module algebra structures. Out of an H-module algebra A one builds the smash product A # H, a single algebra encoding both A and the symmetry H, generalizing the group crossed product and the ring of differential operators.
Let H = U(g) for g = k·d, the one-dimensional Lie algebra, acting on A = k[x] by d = d/dx. Since d is primitive, d · (x · x) = (d · x) x + x (d · x) = 1·x + x·1 = 2x = d · (x²), the Leibniz rule. The smash product A # H is the Weyl algebra.
Differentiation makes k[x] a U(g)-module algebra; the smash product is the Weyl algebra.