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In his book Foundations of quantum group theory, Majid defines (2.1.10) a ribbon Hopf algebra as a quasi-triangular Hopf algebra $(H, R)$ with a special central element $v \in H$ satisfying

(1) $v^2 = u S(u)$ ($u$ being the Drinfeld element),

(2) $\Delta v = (R_{21}R_{12})^{-1}v \otimes v$,

(3) $\varepsilon v =1$,

(4) $Sv=v$.

He claims (without proof) that the axioms are not independent, namely:

Claim 1: (1), (2) imply (4)

Claim 2: (2), (3), (4) imply (1)

I was able neither to prove the claims nor to find some reference where this is done. Can anyone help with a proof/reference?

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  • $\begingroup$ I haven't really checked the proof, but Shum proved that framed tangles form the free ribbon category (whose axioms doesn't have a rule corresponding to (1), only (2-4)). This in particular includes Relation 8 of Fig. 8 from Reshetikhin & Turaev, which is what motivated (1) in the first place. So by following the proofs it should be possible to deduce (2)(3)(4) imply (1). $\endgroup$
    – Trebor
    Commented Nov 11 at 15:23

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