Given a (strict) braided monoidal category $(\mathcal{C},\otimes,I)$ with braiding $b$ and a Hopf algebra $H$ in $\mathcal{C}$. There is a category Rep($H$) of modules over $H$ in $\mathcal{C}$. Do these have a canonical monoidal structure? My candidate for an action on $X\otimes Y$ for $H$-modules $X$ and $Y$ is \begin{equation} \rho_{X\otimes Y}:=(\rho_X\otimes\rho_Y)\circ(1\otimes b_{A,X}\otimes 1)\circ(\Delta\otimes 1\otimes1). \end{equation} The problem is that I can not quite show that it is an action. Perhaps it is only true for symmetric monoidal categories?
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Yes, this is true in a braided monoidal category and your candidate for the action is correct.
A reference is Proposition 2.5 in
Majid, Shahn. Algebras and Hopf algebras in braided categories. Advances in Hopf algebras (Chicago, IL, 1992), 55--105, Lecture Notes in Pure and Appl. Math., 158, Dekker, New York, 1994.
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$\begingroup$ Or look at [Majid, Shahn. Braided groups. J. Pure Appl. Algebra 86 (1993), no. 2, 187--221]. Lemma 1.1. is a comodule version though $\endgroup$ Commented Jul 20, 2017 at 15:28
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1$\begingroup$ Thank you a lot. This is exactly what I was looking for. $\endgroup$– BGJCommented Jul 20, 2017 at 18:32