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In Chapter XVI Kassel introduces a proper definition of a quantum universal enveloping algebra of a Lie algebra $\mathfrak{g}$. (See definition XVI.5.1). Notice that a quantum enveloping algebra has a quasitriangular structure by definition.

Two pages further, Kassel defines a quantum invariant called the canonical $2$-tensor $t$, this definition involves the quasitriangular structure. Is there a categorical interpretation of this element? More precisely, given the category of (finite dimensional) representations of a quantum universal enveloping algebra, can I recover the element $t$ from this braided monoidal category? (The answer is yes if you are also given a fiber functor as you can then recover the Hopf algebra itself, so I'm not interested in that).

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    $\begingroup$ The same definition works on the categorical level. If $\mathcal{C}_\hbar$ is a $k[[\hbar]]$-linear braided monoidal category such that for every pair of objects $\sigma_{x, y}^2 - \mathrm{id}_{x\otimes y}$ is pronilpotent (i.e. divisible by $\hbar$), then you can define $t_{x, y} = (\sigma_{x, y}^2 - \mathrm{id}_{x\otimes y})/\hbar$ which is a symmetric natural endomorphism of $x\otimes y$ in $\mathcal{C}_0$. In this way $\mathcal{C}_0$ becomes what's known as an infinitesimally-braided symmetric monoidal category. $\endgroup$
    – Pavel Safronov
    Dec 22, 2017 at 10:33
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    $\begingroup$ @PavelSafronov: Ah yes that makes sense. I should have read further in Kassel as this is mentioned in chapter XX. If you want you could state this as an answer and I will accept it. $\endgroup$
    – Mathematician 42
    Dec 22, 2017 at 12:04

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