Cluster Nature of Quantum Groups

We present a rigid cluster model to realize the quantum group ${\mathbf{U}}_q(\mathfrak{g})$ for $\mathfrak{g}$ of type ADE. That is, we prove that there is a natural Hopf algebra isomorphism from the quantum group ${\mathbf{U}}_q(\mathfrak{g})$ to a quotient algebra of the Weyl group invariants of the Fock-Goncharov quantum cluster algebra ${\mathcal{O}}_q({\mathcal{P}}_{\mathrm{G},\odot })$. By applying the quantum duality of cluster algebras, we show that ${\mathbf{U}}_q(\mathfrak{g})$ admits a natural basis $\bar{\mathbf{\Theta }}$ whose structural coefficients are in $\mathbb{N}[q^{\frac{1}{2}},q^{-\frac{1}{2}}]$. The basis $\bar{\mathbf{\Theta }}$ satisfies an invariance property under Lusztig’s braid group action, the Dynkin automorphisms, and the star anti-involution.