# Is there a good reference for how ribbon structures change when one switches coproducts?

I'm just going assume readers are familiar with the notions of R-matrix and ribbon categories.

Given a quasi-triangular Hopf algebra $A$ with $R$-matrix $R$, one can construct the co-opposite Hopf algebra with the same multiplication, but the factors switched in the coproduct $\Delta$, and the antipode inverted. Both $R^{21}$ (switched factors) and $R^{-1}$ are $R$-matrix for this new coproduct.

Now, let $v$ be a ribbon element for $(A,R)$. Is $v^{-1}$ a ribbon element for $(A^{op},R^{21})$?

Checking a few examples suggests this is so, and I'm sure I could check it relatively easily. Is there anywhere I can reference?

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It's surely just a short, direct calculation, so I don't know that it's even worth a reference. –  Greg Kuperberg Jan 30 '11 at 9:52