Linearization of Poisson Lie group structures

We show that for any coboundary Poisson Lie group G, the Poisson structure on G^* is linearizable at the group unit. This strengthens a result of Enriquez-Etingof-Marshall, who had established formal linearizability of G^* for quasi-triangular Poisson Lie groups G. We also prove linearizability properties for the group multiplication in G^* and for Poisson Lie group morphisms, with similar assumptions

Linearization of Poisson actions and singular values of matrix products

We prove that the linearization functor from the category of Hamiltonian K-actions with group-valued moment maps in the sense of Lu, to the category of ordinary Hamiltonian K-actions, preserves products up to symplectic isomorphism. As an application, we give a new proof of the Thompson conjecture on singular values of matrix products and extend this result to the case of real matrices. We give a formula for the Liouville volume of these spaces and obtain from it a hyperbolic version of the Duflo isomorphism.Comment: 20 page

Pure Spinors on Lie groups

For any manifold M, the direct sum TM \oplus T*M carries a natural inner product given by the pairing of vectors and covectors. Differential forms on M may be viewed as spinors for the corresponding Clifford bundle, and in particular there is a notion of \emph{pure spinor}. In this paper, we study pure spinors and Dirac structures in the case when M=G is a Lie group with a bi-invariant pseudo-Riemannian metric, e.g. G semi-simple. The applications of our theory include the construction of distinguished volume forms on conjugacy classes in G, and a new approach to the theory of quasi-Hamiltonian G-spaces.Comment: 63 pages. v2: minor changes, typos fixed. To appear in Asterisqu

The Atiyah Algebroid of the Path Fibration over a Lie Group

Let G be a connected Lie group, LG its loop group, and π : PG → G the principal LG-bundle defined by quasi-periodic paths in G. This paper is devoted to differential geometry of the Atiyah algebroid A=T (PG)/LG of this bundle. Given a symmetric bilinear form on $${\mathfrak{g}}$$ and the corresponding central extension of $${L\mathfrak{g}}$$ , we consider the lifting problem for A, and show how the cohomology class of the Cartan 3-form $${\eta \in \Omega^3(G)}$$ arises as an obstruction. This involves the construction of a 2-form $${\varpi \in \Omega^{2}({\rm PG})^{\rm LG}= \Gamma(\wedge^2 A^*)}$$ with $${{\rm d}\varpi=\pi^*\eta}$$ . In the second part of this paper we obtain similar LG-invariant primitives for the higher degree analogues of the form η, and for their G-equivariant extension

Symplectic geometry of Teichm\"uller spaces for surfaces with ideal boundary

A hyperbolic 0-metric on a surface with boundary is a hyperbolic metric on its interior, exhibiting the boundary behavior of the standard metric on the Poincar\'e disk. Consider the infinite-dimensional Teichm\"uller spaces of hyperbolic 0-metrics on oriented surfaces with boundary, up to diffeomorphisms fixing the boundary and homotopic to the identity. We show that these spaces have natural symplectic structures, depending only on the choice of an invariant metric on sl(2,R). We prove that these Teichm\"uller spaces are Hamiltonian Virasoro spaces for the action of the universal cover of the group of diffeomorphisms of the boundary. We give an explicit formula for the Hill potential on the boundary defining the moment map. Furthermore, using Fenchel-Nielsen parameters we prove a Wolpert formula for the symplectic form, leading to global Darboux coordinates on the Teichm\"uller space.Comment: 48 pages, 2 figure

On the coadjoint Virasoro action

The set of coadjoint orbits of the Virasoro algebra at level 1 is in bijection with the set of conjugacy classes in a certain open subset $\widetilde{\rm SL}(2,\mathbb{R})_+$ of the universal cover of ${\rm SL}(2,\mathbb{R})$. We strengthen this bijection to a Morita equivalence of quasi-symplectic groupoids, integrating the Poisson structure on $\mathfrak{vir}^*_\mathsf{1}(S^1)$ and the Cartan-Dirac structure on $\widetilde{\rm SL}(2,\mathbb{R})_+$, respectively.Comment: 40 page