Cluster Nature of Quantum Groups

Time:Fri., 10:00-11:00am, Dec.9, 2022

Venue:Zoom ID: 276 366 7254 ; PW: YMSC

Organizer:Will Donovan, Penghui Li, Peng Shan, Changjian Su

Speaker:Linhui Shen (Michigan State University)


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 $\mathbb{N}[q^{\frac{1}{2}}, q^{-\frac{1}{2}}]$. The basis $\Theta$ satisfies an invariance property under Lusztig's braid group action, the Dynkin automorphisms, and the star anti-involution. Based on a recent preprint arXiv: 2209.06258.


Linhui Shen is an Assistant Professor of Department of Mathematics at Michigan State University.


DATEDecember 9, 2022
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