The Symplectic Geometry of Polygons inEuclidean
نویسندگان
چکیده
We study the symplectic geometry of moduli spaces M r of polygons with xed side lengths in Euclidean space. We show that M r has a natural structure of a complex analytic space and is complex-analytically isomorphic to the weighted quotient of (S 2) n constructed by Deligne and Mostow. We study the Hamiltonian ows on M r obtained by bending the polygon along diagonals and show the group generated by such ows acts transitively on M r. We also relate these ows to the twist ows of Goldman and Jeerey-Weitsman.
منابع مشابه
Polygons in Minkowski Space and Gelfand-tsetlin for Pseudounitary Groups
We study the symplectic geometry of the moduli spaces of polygons in the Minkowski 3-space. These spaces naturally carry completely integrable systems with periodic flows. We extend the Gelfand-Tsetlin method to pseudo-unitary groups and show that the action variables are given by the Minkowski lengths of non-intersecting diagonals.
متن کاملThe Symplectic Geometry of Polygons in the 3-sphere
Abstract. We study the symplectic geometry of the moduli spaces Mr = Mr(S ) of closed n-gons with fixed side-lengths in the 3-sphere. We prove that these moduli spaces have symplectic structures obtained by reduction of the fusion product of n conjugacy classes in SU(2), denoted C r , by the diagonal conjugation action of SU(2). Here C n r is a quasi-Hamiltonian SU(2)-space. An integrable Hamil...
متن کاملA ug 2 00 0 THE SYMPLECTIC GEOMETRY OF POLYGONS IN HYPERBOLIC 3 - SPACE
We study the symplectic geometry of the moduli spaces Mr = Mr(H ) of closed n-gons with fixed side-lengths in hyperbolic three-space. We prove that these moduli spaces have almost canonical symplectic structures. They are the symplectic quotients of B by the dressing action of SU(2) (here B is the subgroup of the Borel subgroup of SL2(C) defined below). We show that the hyperbolic Gauss map set...
متن کاملThe Symplectic Geometry of Polygons in Hyperbolic 3-space∗
We study the symplectic geometry of the moduli spaces Mr = Mr(H) of closed n-gons with fixed side-lengths in hyperbolic three-space. We prove that these moduli spaces have almost canonical symplectic structures. They are the symplectic quotients of Bn by the dressing action of SU(2) (here B is the subgroup of the Borel subgroup of SL2(C) defined below). We show that the hyperbolic Gauss map set...
متن کاملar X iv : m at h / 99 07 14 3 v 1 [ m at h . SG ] 2 2 Ju l 1 99 9 THE SYMPLECTIC GEOMETRY OF POLYGONS IN HYPERBOLIC 3 - SPACE
We study the symplectic geometry of the moduli spaces Mr = Mr(H ) of closed n-gons with fixed side-lengths in hyperbolic three-space. We prove that these moduli spaces have almost canonical symplectic structures. They are the symplectic quotients of B by the dressing action of SU(2) (here B is the standard Borel subgroup of SL2(C)). We show that the hyperbolic Gauss map sets up a real analytic ...
متن کامل