pdf file

Proof.
If H is unimodular, we have α = ɛ for the distinguished group-like element α. Then the
Nakayama involution for the Frobenius structure given by the right cointegral reads by lemma
3.3.9
ρ(h) = 〈α, h (1) 〉S −2 (h (2) ) = 〈ɛ, h (1) 〉S −2 (h (2) ) = S −2 (h) .
For the left integral, one finds ρ(h) = S 2 (h) and thus
Thus
λ(a ∙ b) = λ(b ∙ S 2 (a)) .
(λ ↼ a)(yS 2 x) = λ(ayS 2 x) = λ(xay) .
Thus (λ ↼ a) ∈ C(H), if and only if for all x ∈ H, we have λ(xa) = λ(ax). But this amounts
to a ∈ Z(H).
✷
Theorem 4.4.9 (Drinfeld).
Let (H, R) be a quasi-triangular Hopf algebra with Drinfeld map F R : H ∗ → H. Then
1. For all β ∈ C(H), we have F R (β) ∈ Z(H).
2. For all β ∈ C(H) and α ∈ H ∗ , we have
F R (α ∙ β) = F R (α) ∙ F R (β) .
Proof.
• We calculate for β ∈ C(h) and h ∈ H:
h ∙ F R (β) = h ∙ Q (1) β(Q (2) )
= h (1) Q (1) β(S −1 (h (3) )h (2) Q (2) ) [Lemma 2.5.7]
= Q (1) h (1) β(Q (2) h (2) S(h (3) )) [QΔ = ΔQ and β ∈ C(H)]
= Q (1) β(Q (2) ) ∙ h
= F R (β) ∙ h
• For the second statement, consider α ∈ H ∗ and β ∈ C(H) and calculate
F R (α ∙ β) = R 2 R 1(α ′ ∙ β)(R 1 R 2) ′ [Defn. Drinfeld map]
= R 2 R 1(α ′ ⊗ β)Δ(R 1 R 2) ′ [Defn. product]
= R 2 R 1(α ′ ⊗ β)Δ(R 1 ) ∙ Δ(R 2)
′
= R 2 r 2 s 1 t 1 α(R 1 t 2 )β(r 1 s 2 ) [tensoriality of R, with R = r = s = t]
= R 2 r 2 s 1 β(r 1 s 2 )t 1 α(R 1 t 2 )
= R 2 F R (β)t 1 α(R 1 t 2 ) [Defn. Drinfeld map]
= F R (α) ∙ F R (β) [F R (β) ∈ Z(H)]
104