2022/09/13 by Linhui Shen, Shen, Linhui
Mathematics · Chemistry · #Algebraic structures and combinatorial models #Advanced Algebra and Geometry #Molecular spectroscopy and chirality
paper · pdf · doi:10.48550/arxiv.2209.06258
We present a rigid cluster model to realize the quantum group \bf Uq(\mathfrakg) for \mathfrakg of type ADE. That is, we prove that there is a natural Hopf algebra isomorphism from the quantum group \bf Uq(\mathfrakg) to a quotient algebra of the Weyl group invariants of the Fock-Goncharov quantum cluster algebra Oq(\mathscrP_\rm G,\odot). By applying the quantum duality of cluster algebras, we show that \bf Uq(\mathfrakg) admits a natural basis \bf Θ whose structural coefficients are in ℕ[q(1)/(2), q-(1)/(2)]. The basis \bf Θ satisfies an invariance property under Lusztig's braid group action, the Dynkin automorphisms, and the star anti-involution.