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In the proposition below we summarize the main properties of the antipode in a Hopf algebra:

PROPOSITION 1.4.– (see Sweedler (1969, Proposition 4.0.1)). Let ℋ be a Hopf algebra with multiplication m, comultiplication Δ, unit u : 1 ↦ 1, counit ε and antipode S. Then:

1 1) S ∘ u = u and ε o S = ε.

2 2) S is an algebra antimorphism and a coalgebra antimorphism, that is, if denotes the flip, we have:

3 3) If ℋ is commutative or cocommutative, then S2 = I.

For a detailed proof, see Kassel (1995).

PROPOSITION 1.5.–

1 1) If x is a primitive element, then S(x) = –x.

2 2) The linear subspace Prim ℋ of primitive elements in ℋ is a Lie algebra.

PROOF.– If x is primitive, then (ε ⊗ ε) ∘ Δ(x) = 2ε(x). On the contrary, (ε ⊗ ε) ∘ Δ(x) = ε(x), so ε(x) = 0. Then:

Now let x and y be the primitive elements of ℋ. Then, we can easily compute: