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In a nutshell, my question is:

Q0: is there a classification of invariant of (framed) tangles arising from the Reshetikhin–Turaev construction?

I will now make it more precise. One could define a functorial invariant of (framed) tangles to be a ribbon monoidal functor $$ F\colon\mathbf{FrTang}\to \mathcal{V}, $$ where $\mathbf{FrTang}$ is the ribbon monoidal category (or tortile category) of oriented framed tangles and $\mathcal{V}$ is a ribbon monoidal category. By a result of Shum, $\mathbf{FrTang}$ is the free ribbon monoidal category on one object, so that such functor is determined by a choice of object in $\mathcal{V}$. Hence, classification reduces to classification of ribbon monoidal categories.

On the other hand, the Reshetikhin–Turaev construction states that given a ribbon Hopf algebra $A$, its category of finite-dimensional representations $\mathrm{mod}(A)$ is a ribbon monoidal category.

This leads me to the following two questions:

Q1: Does the Reshetikhin–Turaev construction give all ribbon monoidal categories, and does $\mathrm{mod}(A)$ determines $A$, taking the ribbon structure and fiber functor $\mathrm{mod}(A)\to\mathrm{vec}$ into account?

In other words, is there a Tannakian duality for ribbon monoidal categories, akin to the one for rigid (or autonomous) braided monoidal categories and quasi-triangular Hopf algebras?

Assuming a positive (or at least optimistic) answer to Q1:

Q2: is there a classification of ribbon Hopf algebras? Or at least of those of the form $U_q(\mathfrak{g})$ for some Lie algebra $\mathfrak{g}$?

Any partial answer would be greatly appreciated.

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Q1 asks two very different questions: the answer to the first one is definitely not, and the answer to the second one is almost.

Let me start with the second one. First of all, to handle infinite dimensional examples like $U_q(\mathfrak g)$ (which, strictly speaking, is not a ribbon Hopf algebra), as usual with Tannaka duality it's better to consider instead (finite-dimensional) comodules over a co-ribbon Hopf algebra. Then the general slogan is that the "hard" part is to reconstruct $A$ as a bialgebra, and then the extra structures or properties (Hopf, braided, ribbon,..) come pretty much for free. Still, knowing the underlying monoidal category is not enough, you also need to have the monoidal forgetful functor to vect.

So at the end of the day I'd argue your question has little to do with the ribbon structure, and basically boils down to whether all monoidal categories come from bialgebras, which is well-known to be false. The most obvious example in the ribbon case is of course $FrTang$ itself. There are also lots of examples which are full sub-categories of ribbon categories of the form you want, without being of this form themselves. Even if you do restrict to categories which you know are categories of (co)modules, there are lots of weakenings of the axioms of Hopf algebras (e.g. quasi Hopf, weak Hopf, Hopf algebroids, etc..), and in all of theses cases it makes sense to have a ribbon structure.

For the second question, if $q=e^h$ for a formal variable $h$ (which is the only thing that makes sense for arbitrary Lie algebra) then those are essentially classified by formal solutions $r \in \mathfrak g^{\otimes 2}[[h]]$ of the classical Yang-Baxter equation, for which $r+r^{2,1}$ is $\mathfrak g$-invariant. If you want $q$ to be a variable, there are some known results for reductive Lie algebras, see e.g. https://mathoverflow.net/a/314487/13552.

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  • $\begingroup$ Thank you for your answer. Q1. To clarify, I definitely mean "determine" as taking the ribbon structure and fiber functor into account (I edited the question). Hence at least the quasi-triangular Hopf structure is determined. Should I then conclude that the additional ribbon structure on the algebra is determined by the ribbon structure on the category? $\endgroup$
    – Léo S.
    Commented May 20 at 17:10
  • $\begingroup$ Yes, an endomorphism of the identity functor is the same as an element in the center of $A$. There is nothing special about ribbon or quasi-triangular, once you know your category if $mod(A)$ for some f.d. bialgebra, basically every structure or property you can think of can be recovered this way (as long as it makes sense categorically). $\endgroup$
    – Adrien
    Commented May 20 at 19:16
  • $\begingroup$ Thank you, I now understand the point you were making. Regarding Q2, I guess a summary could be that (i) for general ribbon Hopf algebras (not necessarily viewed as deformations), there is nothing near a classification (and maybe no hope of ever having one, even with some restrictions eg f.d.?); (ii) for ribbon Hopf algebras arising from formal deformations, we are possibly as close as one could be to a classification; and (iii) for non-formal deformations, there are results in classical types ABCD (and G2), but work remains to be done (and progress seems possible) $\endgroup$
    – Léo S.
    Commented May 21 at 7:26
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    $\begingroup$ Yes, again forgetting about ribbon there are partial results in the literature on classification of f.d., even semi-simple, Hopf algebras and this is a very hard problem: e.g. this includes the classification of finite groups (which tautologically lead to semi-simple f.d. ribbon Hopf algebras) which is already extremely difficult. In a way a more natural problem would be to classify finite tensor (or ribbon), possibly fusion, categories instead, and again there are lots of partial result, but AFAIK nothing near a complete classification. $\endgroup$
    – Adrien
    Commented May 21 at 7:40
  • $\begingroup$ Okay, thanks! This clarifies things a lot $\endgroup$
    – Léo S.
    Commented May 21 at 9:46

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