$\newcommand{\g}{\mathfrak g}$ I think the statement Scott Carnahan was refeering to in his answer concerns in fact formal deformations of representations of $\g$, i.e. deformations over the ring $\mathbb C[[\hbar]]$. In that case the reference is Drinfeld's "On quasitriangular quasi-Hopf algebras and on a group that is closely connected with $Gal(\bar{\mathbb Q}/\mathbb Q)$". If I remember correctly, it says more generally that for any finite-dimensional Lie algebra there is a bijection between equivalence classes of such deformations as monoidal and braided monoidal categories, and $(\wedge^3 \g)^\g[[\hbar]]$ and $S^2(\g)^\g[[\hbar]]$ respectively. Note that there is a map from the latter to the former given by $$c\mapsto [c^{1,2},c^{2,3}],$$ which is the infinitesimal analog of forgetting the braiding. Now, if $\g$ is semi-simple those spaces are one dimensional (the second one is spanned by the inverse of the Killing form), and the previous map is an isomorphism.

There is a nice interpretation of this and of similar results in the framework of shifted Poisson geometry, I highly recommend having a look at Pavel Safronov's very cool paper "Poisson-Lie structures as shifted Poisson structures" (arXiv:1706.02623).

Edit: On second thoughts Drinfeld actually shows this result for braided deformations in general, and I think also for monoidal ones in the semi-simple case. The general results for monoidal deformations is true but more recent.

$\newcommand{\g}{\mathfrak g}$ I think the statement Scott Carnahan was refeering to in his answer concerns in fact formal deformations of representations of $\g$, i.e. deformations over the ring $\mathbb C[[\hbar]]$. In that case the reference is Drinfeld's "On quasitriangular quasi-Hopf algebras and on a group that is closely connected with $Gal(\bar{\mathbb Q}/\mathbb Q)$". If I remember correctly, it says more generally that for any finite-dimensional Lie algebra there is a bijection between equivalence classes of such deformations as braided monoidal categories and $S^2(\g)^\g[[\hbar]]$, and if $\g$ is simple this is one dimensional. I think his results also implies that in the simple case those are the only deformations as monoidal categories as well (this is definitevely true, I'm just unsure if it can be extracted from his paper directly).

There is a nice interpretation of this and of similar results in the framework of shifted Poisson geometry, I highly recommend having a look at Pavel Safronov's very cool paper "Poisson-Lie structures as shifted Poisson structures" (arXiv:1706.02623). In particular this basically settle the question for monoidal deformations and arbitrary finite-dimensional $\g$.

Edit: My first answer was claiming something wrong in the general case, I corrected.

$\newcommand{\g}{\mathfrak g}$ I think the statement Scott Carnahan was refeering to in his answer concerns in fact formal deformations of representations of $\g$, i.e. deformations over the ring $\mathbb C[[\hbar]]$. In that case the reference is Drinfeld's "On quasitriangular quasi-Hopf algebras and on a group that is closely connected with $Gal(\bar{\mathbb Q}/\mathbb Q)$". If I remember correctly, it says more generally that for any finite-dimensional Lie algebra there is a bijection between equivalence classes of such deformations as monoidal and braided monoidal categories, and $(\wedge^3 \g)^\g[[\hbar]]$ and $S^2(\g)^\g[[\hbar]]$ respectively. Note that there is a map from the latter to the former given by $$c\mapsto [c^{1,2},c^{2,3}],$$ which is the infinitesimal analog of forgetting the braiding. Now, if $\g$ is semi-simple those spaces are one dimensional (the second one is spanned by the inverse of the Killing form), and the previous map is an isomorphism.

There is a nice interpretation of this and of similar results in the framework of shifted Poisson geometry, I highly recommend having a look at Pavel Safronov's very cool paper "Poisson-Lie structures as shifted Poisson structures" (arXiv:1706.02623).

$\newcommand{\g}{\mathfrak g}$ I think the statement Scott Carnahan was refeering to in his answer concerns in fact formal deformations of representations of $\g$, i.e. deformations over the ring $\mathbb C[[\hbar]]$. In that case the reference is Drinfeld's "On quasitriangular quasi-Hopf algebras and on a group that is closely connected with $Gal(\bar{\mathbb Q}/\mathbb Q)$". If I remember correctly, it says more generally that for any finite-dimensional Lie algebra there is a bijection between equivalence classes of such deformations as monoidal and braided monoidal categories, and $(\wedge^3 \g)^\g[[\hbar]]$ and $S^2(\g)^\g[[\hbar]]$ respectively. Note that there is a map from the latter to the former given by $$c\mapsto [c^{1,2},c^{2,3}],$$ which is the infinitesimal analog of forgetting the braiding. Now, if $\g$ is semi-simple those spaces are one dimensional (the second one is spanned by the inverse of the Killing form), and the previous map is an isomorphism.

There is a nice interpretation of this and of similar results in the framework of shifted Poisson geometry, I highly recommend having a look at Pavel Safronov's very cool paper "Poisson-Lie structures as shifted Poisson structures" (arXiv:1706.02623).

Edit: On second thoughts Drinfeld actually shows this result for braided deformations in general, and I think also for monoidal ones in the semi-simple case. The general results for monoidal deformations is true but more recent.