Linhui Shen: Cluster Nature of Quantum Groups
We present a rigid cluster model to realize the quantum group U_q(g) for g of type ADE. That is, we prove that there is a natural Hopf algebra isomorphism from the quantum group to a quotient algebra of the Weyl group invariants of a Fock-Goncharov quantum cluster algebra. By applying the quantum duality of cluster algebras, we show that the quantum group admits a cluster canonical basis Theta whose structural coefficients are in Laurent polynomials with non negative integer coefficients in the square root of q. The basis Theta satisfies an invariance property under Lusztig's braid group action, the Dynkin automorphisms, and the star anti-involution.