Poisson-Lie anomalies

Abstract:
We describe the concept of (nonlinear) Poisson-Lie cohomology of Lie algebras and show that it classifies possible anomalies of Poisson-Lie symmetries. Given a Poisson-Lie 2-cocycle, we also explicitly construct a symplectic manifold which realizes the corresponding anomaly. We call this symplectic manifold an anomalous Heisenberg double since it can be viewed as a generalisation of the Heisenberg double of Semenov-Tian-Shansky. We show that particular classes of Poisson-Lie anomalies naturally arise by taking a limit q®¥ of known Poisson-Lie symmetric dynamical systems. In particular, we establish that the q®¥ limit of the quasitriangular q-WZW model is underlied by a particular anomalous Heisenberg double of the loop group.