A Poisson-Lie group $G$ with its standard Poisson structure admits a family of cluster coordinates with the defining property of having log-canonical Poisson brackets:
$X_i, X_j = a_{ij} X_i X_j$. On the level of quantum groups, these coordinates become a family of $q$-commuting generators $X_i X_j = q^{a_{ij}} X_j X_i$ for (a localization of) the quantized algebra $O_q[G]$ of functions on $G$.
Showing that a localization of the quantum group $U_q(g)$ is isomorphic to certain quantum torus algebra (i.e. an algebra with $q$-commuting generators) is a much desired property known as the quantum Gelfand-Kirillov conjecture. In a joint work with Gus Schrader, we have constructed an embedding of the quantum group $U_q(g)$ into a quantum torus algebra naturally defined from the quantum double Bruhat cell $O_q[G^{w_0,w_0}]$. Our construction is motivated by Poisson geometry of the Grothendieck-Springer resolution and is closely related to the global sections functor of the quantum Beilinson-Bernstein theorem. I will explain our work, outline a few other ways to obtain such cluster structure on $U_q(g)$, and discuss its applications to representation theory.