dual Hopf algebra

A Hopf algebra has its structure maps split evenly between 'algebra' arrows and 'coalgebra' arrows, and dualizing — flipping every linear map around — swaps the two halves while preserving the overall shape. The dual of a Hopf algebra is therefore again a Hopf algebra, with multiplication and comultiplication trading places. This perfect symmetry is one of the most elegant features of the theory and underlies the duality between 'functions on a group' and 'the group algebra.'

For a finite-dimensional Hopf algebra H, the linear dual H* = Hom(H, k) is a Hopf algebra with multiplication dual to Δ_H, comultiplication dual to m_H, unit dual to ε_H, counit dual to u_H, and antipode S_{H*} = (S_H)*. The pairing ⟨f g, h⟩ = Σ f(h(1)) g(h(2)) and ⟨Δ(f), h ⊗ h'⟩ = f(h h') make this explicit. In infinite dimensions one replaces H* by the restricted (Hopf) dual H°, the largest subspace whose dual structure closes up.

The two prototypes are dual to each other: the group algebra k[G] of a finite group and the function algebra k^G of functions on G are dual Hopf algebras. The cocommutative side (k[G]) dualizes to the commutative side (k^G) and vice versa. For quantum groups this duality pairs the deformed coordinate algebra O_q(G) with the deformed enveloping algebra U_q(g), so studying one is studying the other.