# Yetter--Drinfeld Modules and Braidings

Let $H$ be a Hopf algebra, $M$ a right $H$-comodule for a coaction $\Delta_R$, and $\triangleleft$ a right $H$-action on $M$ such that $M$ is a Yetter--Drindfeld module. We know from general theory that we have a corresponding braiding $$\sigma: M \to M, ~~~~~~~~~~ m \otimes n \mapsto n_{(0)} \otimes (m \triangleleft n_{(1)}),$$ where $n_{(0)} \otimes n_{(1)}$ is the image of $n$ under $\Delta_R$.

Questions: (1) Can there exist another right action giving $M$ (still endowed with $\Delta_R$) the structure of a Yetter--Drinfeld module and producing the same braiding?

(2) Can there exist another right coaction giving $M$ (still endowed with $\triangleleft$) the structure of a Yetter--Drinfeld module and producing the same braiding?

(3) Do the answers to these questions change if $M$ is assumed to be finite dimensional?

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There are nonisomorpihic modules that have isomorphic $H$ and $H^*$ actions (and therefore isomorphic braidings?)