Cluster Nature of Quantum Groups

Abstract

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