Geometry of the analytic loop group

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 U∈(ĝ) 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 U∈(ĝ) (it is a geometric realization of the center). Then the symplectic leaves, and corresponding equivalence classes of central characters of U∈(ĝ), are parameterized by certain G-bundles on an elliptic curve. © 2013 Elsevier Ltd.