Non-coboundary Poisson-Lie structures on the book group

All possible Poisson-Lie (PL) structures on the 3D real Lie group generated by a dilation and two commuting translations are obtained. Its classification is fully performed by relating these PL groups with the corresponding Lie bialgebra structures on the corresponding "book" Lie algebra. By construction, all these Poisson structures are quadratic Poisson-Hopf algebras for which the group multiplication is a Poisson map. In contrast to the case of simple Lie groups, it turns out that most of the PL structures on the book group are non-coboundary ones. Moreover, from the viewpoint of Poisson dynamics, the most interesting PL book structures are just some of these non-coboundaries, which are explicitly analysed. In particular, we show that the two different q-deformed Poisson versions of the sl(2,R) algebra appear as two distinguished cases in this classification, as well as the quadratic Poisson structure that underlies the integrability of a large class of 3D Lotka-Volterra equations. Finally, the quantization problem for these PL groups is sketched.Comment: 15 pages, revised version, some references adde

Dual generators of the fundamental group and the moduli space of flat connections

We define the dual of a set of generators of the fundamental group of an oriented two-surface $S_{g,n}$ of genus $g$ with $n$ punctures and the associated surface $S_{g,n}\setminus D$ with a disc $D$ removed. This dual is another set of generators related to the original generators via an involution and has the properties of a dual graph. In particular, it provides an algebraic prescription for determining the intersection points of a curve representing a general element of the fundamental group $\pi_1(S_{g,n}\setminus D)$ with the representatives of the generators and the order in which these intersection points occur on the generators.We apply this dual to the moduli space of flat connections on $S_{g,n}$ and show that when expressed in terms both, the holonomies along a set of generators and their duals, the Poisson structure on the moduli space takes a particularly simple form. Using this description of the Poisson structure, we derive explicit expressions for the Poisson brackets of general Wilson loop observables associated to closed, embedded curves on the surface and determine the associated flows on phase space. We demonstrate that the observables constructed from the pairing in the Chern-Simons action generate of infinitesimal Dehn twists and show that the mapping class group acts by Poisson isomorphisms.Comment: 54 pages, 13 .eps figure

Boundary conditions and symplectic structure in the Chern-Simons formulation of (2+1)-dimensional gravity

We propose a description of open universes in the Chern-Simons formulation of (2+1)-dimensional gravity where spatial infinity is implemented as a puncture. At this puncture, additional variables are introduced which lie in the cotangent bundle of the Poincar\'e group, and coupled minimally to the Chern-Simons gauge field. We apply this description of spatial infinity to open universes of general genus and with an arbitrary number of massive spinning particles. Using results of [9] we give a finite dimensional description of the phase space and determine its symplectic structure. In the special case of a genus zero universe with spinless particles, we compare our result to the symplectic structure computed by Matschull in the metric formulation of (2+1)-dimensional gravity. We comment on the quantisation of the phase space and derive a quantisation condition for the total mass and spin of an open universe.Comment: 44 pages, 3 eps figure