Geometry of the analytic loop group

TL;DR

This paper develops the theory of an analytic loop group with a Riemann-Hilbert factorization, linking it to quantum affine algebras at roots of unity, and explores its Poisson structure and geometric properties.

Contribution

It introduces a new notion of analytic loop group with a Riemann-Hilbert factorization and connects its Poisson dual to the center of quantum affine algebras at roots of unity.

Findings

Poisson structure is isomorphic to the semi-classical limit of the quantum affine algebra center

Symplectic leaves correspond to G-bundles on elliptic curves

Provides a geometric realization of the quantum affine algebra center

Abstract

We introduce and study a notion of analytic loop group with a Riemann-Hilbert factorization relevant for the representation theory of quantum affine algebras at roots of unity with non trivial central charge. We introduce a Poisson structure and study properties of its Poisson dual group. We prove that the Hopf-Poisson structure is isomorphic to the semi-classical limit of the center of the quantum affine algebra (it is a geometric realization of the center). Then the symplectic leaves, and corresponding equivalence classes of central characters, are parameterized by certain G-bundles on an elliptic curve.