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A braided vector space is a pair $(V,\sigma)$ consisting of a vector space $V$, and a linear map $\sigma:V \otimes V \to V \otimes V$, satisfying the Yang--Baxter equation. Ee can scale the braiding by $\lambda \in \mathbb{C}$ to produce a new braiding $\lambda \sigma$.

Given a Yetter--Drinfeld module $(V,\bullet,\delta)$, a braiding is given by $$ \sigma: V \otimes V \to V \otimes V, ~~~~~~~ v \otimes w \mapsto v_{(-1)}\bullet w \otimes v_{(0)}. $$ As above, scaling this braiding again gives a braiding - however it does not come from any obvious rescaling of the Yetter--Drinfeld module. Is their some clever way to scale $(V,\bullet,\delta)$ so that its asociated braiding is $\lambda \sigma$?

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    $\begingroup$ You have to rescale the underlying Hopf algebra $H$ for this. Take $\lambda\Delta$, $\lambda^{-1}\epsilon$ and $\lambda\delta$. $\endgroup$
    – Christoph
    Apr 7, 2019 at 12:25

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