The defining equation
A Hopf algebra is a bialgebra H equipped with a linear map S: H → H, the antipode, satisfying one equation. Recall from the last guide that End(H) is an algebra under convolution *, with unit u∘ε. The antipode axiom is simply that S is a two-sided *-inverse of the identity map: S * id = id * S = u∘ε. In Sweedler notation this reads S(h₍₁₎) h₍₂₎ = ε(h) 1 = h₍₁₎ S(h₍₂₎). That is the whole definition — one short equation, but a remarkably powerful one.
Computing S on the two examples
On the group algebra k[G], every g is grouplike, Δ(g) = g⊗g, ε(g)=1. The antipode axiom S(g)·g = ε(g)1 = 1 forces S(g) = g⁻¹. The antipode is literally group inversion, extended linearly. This is the cleanest reason to think of the antipode as a generalized inverse. On the universal enveloping algebra U(g) of a Lie algebra g, the generators x∈g are primitive: Δ(x) = x⊗1 + 1⊗x, ε(x)=0. The axiom now gives S(x) = −x: on primitives the antipode is negation, the algebraic shadow of inverting a one-parameter subgroup exp(tx) to exp(−tx).
Antipode on a primitive element x: Delta(x) = x (x) 1 + 1 (x) x, eps(x)=0.
Demand S(x_(1)) x_(2) = eps(x) 1 = 0.
Expand the left side over the two Sweedler terms of Delta(x):
S(x).1 + S(1).x = 0.
Since S(1) = 1 (antipode fixes the unit), this is
S(x) + x = 0 => S(x) = -x. QED.
Antipode is an ANTI-homomorphism: S(ab) = S(b) S(a).
Check on U(g) with primitives x, y:
S(xy) should equal S(y)S(x) = (-y)(-x) = yx.
Also S(yx) = S(x)S(y) = xy.
Difference: S(xy) - S(yx) = yx - xy = -[x,y] ... wait, recompute:
S(xy - yx) = S([x,y]). Since [x,y] is primitive,
S([x,y]) = -[x,y] = -(xy - yx) = yx - xy. Consistent. OK.
So S reverses order, matching (ab)^{-1} = b^{-1} a^{-1} in groups.Properties that fall out for free
From the single axiom you can derive, by clean convolution arguments, that S is an algebra anti-homomorphism — S(ab) = S(b)S(a), S(1) = 1 — and a coalgebra anti-homomorphism. The order reversal mirrors (ab)⁻¹ = b⁻¹a⁻¹. Be honest about one subtlety: S need not be its own inverse. In commutative or cocommutative Hopf algebras S² = id, but in general — and crucially in the quantum groups of Guide 5 — S² is a nontrivial automorphism. That failure of S² = id is one of the first signs that quantum groups are not just groups in disguise.