antipode

In a group, every element has an inverse. The antipode is what 'taking inverses' becomes once you replace the group by a Hopf algebra. It is a linear map S that, when combined with the splitting-and-multiplying machinery of the coproduct, cancels everything down to the counit — exactly as multiplying an element by its inverse collapses to the identity. So the antipode is the Hopf-algebraic shadow of inversion.

Precisely, the antipode of a Hopf algebra H is the two-sided convolution inverse of the identity map id_H in the convolution algebra (Hom(H, H), *, u∘ε). That is, S satisfies S * id = u ∘ ε = id * S, which in Sweedler notation reads Σ S(c(1)) c(2) = ε(c) 1 = Σ c(1) S(c(2)). Because convolution inverses are unique when they exist, the antipode is uniquely determined by the bialgebra structure.

The antipode is always an anti-homomorphism for both structures: S(ab) = S(b)S(a) and (S ⊗ S) ∘ Δ = τ ∘ Δ ∘ S, where τ is the flip. It need not be invertible as a linear map, though S² = id whenever H is commutative or cocommutative. In group algebras S(g) = g^{-1}, so S² = id; in genuine quantum groups S² is a nontrivial inner-type automorphism, which is one symptom of their non(co)commutativity.