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We need to understand the powers of the antipode. We first need another structure: for any
element h ∈ H, left multiplication yields a K-linear endomorphism
We thus define a linear form
L h : H → H
x ↦→ hx
Tr H : H → K
h ↦→ Tr H (L h ) .
Proposition 3.3.3.
Let H be a finite-dimensional Hopf algebra with λ ∈ I l (H ∗ ) and a right integral Λ ∈ H such
that λ(Λ) = 1.
1. We have
TrS 2 = 〈ɛ, Λ〉〈λ, 1〉 .
2. If S 2 = id H , then Tr H = 〈ɛ, Λ〉λ.
Proof.
1. Taking S 2 in lemma 3.3.2, we find
Tr(S 2 ) = 〈λ, S 2 (Λ (2) )S(Λ (1) )〉 = 〈λ, S(Λ (1) ∙ S(Λ (2) ))〉 = 〈ɛ, Λ〉 ∙ 〈λ, 1〉 .
2. The identity S 2 = id H implies by proposition 2.5.6
h (2) S(h (1) ) = 〈ɛ, h〉1 for all h ∈ H .
Taking F = L h , we find
Tr H (h) = Tr H (L h ) = 〈λ, hΛ (2) S(Λ (1) )〉 = 〈ɛ, Λ〉 ∙ 〈λ, h〉 ,
where we used in the last step the previous equation for h = Λ.
✷
Corollary 3.3.4.
1. H and H ∗ are semisimple, if and only if TrS 2 ≠ 0.
2. If S 2 = id H and charK does not divide dim H, then H and H ∗ are semisimple.
Proof.
1. By Maschke’s theorem 3.2.13, H is semisimple, if and only if 〈ɛ, Λ〉 ̸= 0. Similarly, again
by Maschke’s theorem, H ∗ is semisimple, if and only if 〈ɛ ∗ , Λ ∗ 〉 = 〈λ, 1〉 ̸= 0. Together
with proposition 3.3.3.1, this implies the assertion.
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