# Scaling Yetter--Drinfeld Modules

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 $$\lambda \sigma$$?

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