Fusion and specialization for type ADE shuffle algebras

TL;DR

This paper constructs fused currents in type ADE quantum loop groups as duals to specialization maps in shuffle algebras, highlighting their dependence on convex root orders and using the Auslander-Reiten order.

Contribution

It introduces a new construction of fused currents in ADE types as duals to shuffle algebra specialization maps, based on a specific convex root order.

Findings

Fused currents depend on convex root orders.

Construction uses the Auslander-Reiten order.

Establishes duality between fused currents and specialization maps.

Abstract

Root vectors in quantum groups (of finite type) generalize to fused currents in quantum loop groups ([5]). In the present paper, we construct fused currents as duals to specialization maps of the corresponding shuffle algebras ([7,8,9]) in types ADE. Both root vectors and fused currents depend on a convex order of the positive roots, and the choice we make in the present paper is that of the Auslander-Reiten order ([24]) corresponding to an orientation of the type ADE Dynkin diagram.