The dual swaps multiplication and comultiplication
Let H be a finite-dimensional Hopf algebra and H* = Hom(H, k) its dual space. Dualizing each structure map and using (H⊗H)* ≅ H*⊗H* turns H* into a Hopf algebra. The multiplication on H* is the transpose of Δ: (φ·ψ)(h) = (φ⊗ψ)(Δh) = φ(h₍₁₎)ψ(h₍₂₎) — note this is exactly the convolution of φ and ψ into k. The comultiplication on H* is the transpose of m. The counit of H* is evaluation at 1; the unit is ε; the antipode is S*. In one sentence: the dual Hopf algebra is the same data with the roles of m and Δ interchanged.
A finite group, seen twice
Take G finite. Its group algebra k[G] is a Hopf algebra with grouplike basis. Its dual is O(G), the algebra of k-valued functions on G under pointwise multiplication, with dual basis {δ_g} of indicator functions. Watch the roles swap. In k[G] the product is the group law g·h; dualizing, the comultiplication on O(G) is Δ(δ_x) = Σ_{g h = x} δ_g ⊗ δ_h — it remembers the group law. Meanwhile the product in O(G) is pointwise, δ_g δ_h = δ_{g,h} δ_g, which dualizes to the grouplike Δ(g)=g⊗g of k[G]. The group law and the diagonal have traded sides.
G = Z/3Z = {0, 1, 2}, additive group. Two dual Hopf algebras.
(1) Group algebra k[G], basis e_0, e_1, e_2 (grouplike):
e_i . e_j = e_{i+j mod 3} (twisted convolution product)
Delta(e_i) = e_i (x) e_i, eps(e_i) = 1, S(e_i) = e_{-i}.
(2) Functions O(G), basis d_0, d_1, d_2 (indicator functions):
d_i . d_j = delta_{ij} d_i (pointwise product, idempotents)
Delta(d_k) = sum_{i+j = k mod 3} d_i (x) d_j, eps(d_k)=delta_{k,0},
S(d_k) = d_{-k}.
Pairing <d_i, e_j> = delta_{ij} identifies O(G) = k[G]*.
Check the swap once: the product e_1 . e_1 = e_2 in k[G]
corresponds, under transpose, to the coefficient of d_1 (x) d_1
in Delta(d_2) -- and indeed 1+1 = 2, so d_1 (x) d_1 appears. Consistent.
Note: for abelian G these two Hopf algebras are isomorphic via the
characters (Pontryagin / Fourier duality). For nonabelian G they differ:
k[G] is cocommutative but noncommutative; O(G) is commutative but not
cocommutative.Comodules: dualizing the action
Dualize the notion of module too. A right comodule over a coalgebra C is a vector space V with a coaction ρ: V → V⊗C satisfying coassociativity and counit conditions mirroring the module axioms. Over O(G) a comodule is the same as a G-graded vector space; over k[G] a comodule is the same as a representation. So the module/comodule duality interchanges representations of G with G-gradings — concrete, useful, and the template for how quantum-group comodules will encode braided structure in the final guide.