Hall algebras and quantum groups associated to Dynkin quivers

TL;DR

This paper constructs Hall algebras and quantum groups for Dynkin quivers using explicit Laurent polynomials, extending the understanding of their algebraic structures and providing explicit formulas in specific cases.

Contribution

It introduces a new construction of Hall algebras for Dynkin quivers via Laurent polynomials and derives full quantum groups for arbitrary parameters.

Findings

Explicit Laurent polynomials for Dynkin quivers.

Construction of Hall algebras over Laurent polynomial rings.

Derivation of full quantum groups for arbitrary parameters.

Abstract

For Dynkin quivers, we find the Laurent polynomials $\widetilde{X}_{a, c}^{b}(v)$ and use $\widetilde{X}_{a, c}^{b}(v)$ to construct the Hall algebra $\hc_v(\cc(\cp))$ over $\mz[v, v^{-1}]$, where $\widetilde{X}_{a, c}^{b}(|\mf_q|)$'s are structure constants used by Bridgeland. The Laurent polynomials $\widetilde{X}_{a, c}^{b}(v)$ are explicitly given in $A_1$ case. As an application, we obtain the full quantum groups $U_t(\sg)$ associated to the Dynkin quivers for arbitrary $t\not=0,\pm1$.