Quantum loop groups for symmetric Cartan matrices
Andrei Neguţ
TL;DR
This work constructs quantum loop groups for general symmetric Cartan matrices by a Drinfeld-double framework, introducing the radical relations $I^{\pm}$ via a perfect Hopf pairing and expressing the positive/negative halves as quotients $U_q^{\pm}(L\mathfrak{g})$. Central to the approach is a shuffle-algebra model: symmetric rational functions $\mathcal{V}^{\pm}$ with wheel constraints define subalgebras $\mathcal{S}^{\pm}$ whose images under $\widetilde{\Upsilon}^{\pm}$ realize $U_q^{\pm}(L\mathfrak{g})$, yielding a nondegenerate pairing with their duals. Distinguished zig-zags and refined selections encode the loop-Serre-type relations via $\rho_Z$, and the authors prove $U_q^{\pm}(L\mathfrak{g})\cong \mathcal{S}^{\pm}$, establishing a concrete, combinatorial presentation of quantum loop groups. The paper then connects these algebraic structures to $K$-theoretic Hall algebras of quivers without loops, showing that the localized nilpotent KHA $K^{\text{nilp}}_{\mathbb{C}^*,\text{loc}}$ realizes the quantum loop group in simply-laced types, with a geometry-adapted shuffle algebra $\mathcal{S}^{+\text{,geom}}$ governing the correspondence. Together, these results provide a unified framework linking quantum loop groups, shuffle algebras, and geometric representation theory via $K$-theoretic Hall algebras.
Abstract
We introduce a quantum loop group associated to a general symmetric Cartan matrix, by imposing just enough relations between the usual generators $\{e_{i,k}, f_{i,k}\}_{i \in I, k \in \mathbb{Z}}$ in order for the natural Hopf pairing between the positive and negative halves of the quantum loop group to be perfect. As an application, we describe the localized K-theoretic Hall algebra of any quiver without loops, endowed with a particularly important $\mathbb{C}^*$ action.