Search arXiv⌕ Search

arXiv · math/0208051

A Poisson structure on compact symmetric spaces

Abstract

We present some basic results on a natural Poisson structure on any compact symmetric space. The symplectic leaves of this structure are related to the orbits of the corresponding real semisimple group on the complex flag manifold.

Explore related subjects

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Philip Foth, Jiang-Hua Lu. 2003-09-24. A Poisson structure on compact symmetric spaces. https://arxiv.org/abs/math/0208051

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

Algebra of the Infrared with Elliptic Curve-Valued Potential

We study an extension of the algebra of the infrared to curve--valued potentials, focusing on the elliptic curve case. Given a finite configuration of points on an elliptic curve, we construct associated \(L_\infty\)-algebra. Adding an extra point produces a directed \(A_\infty\)-algebra and an \(L_\infty\)-morphism to their directed Hochschild deformation complexes. Maurer--Cartan elements of the \(L_\infty\)-algebra determine \(A_\infty\)-deformations of the directed algebra. We formulate a conjectural comparison between Maurer--Cartan deformations of the directed algebra and the directed Fukaya--Seidel category associated to compatible finite collections of thimbles.

math.SG↗

Moment Lagrangians, unobstructedness and symplectic groupoids

Moment Lagrangian $L_μ$ is a Lagrangian in $T^*G^- \times Y^- \times Y$ associated to a Hamiltonian $G$-space $Y$ with a moment map $μ$. In this paper, we prove that $L_μ$ is tautologically unobstructed under mild assumptions on $Y$. As a key ingredient in the proof, we constructed a new symplectic groupoid structure on $T^*G^- \times Y^- \times Y$ over $G\times Y$ for which $L_μ$ is simultaneously the unit and the fixed locus of the inversion, which might be of independent interest.

math.SG↗

The symplectic leaves of the affine Poisson $C^\infty$-scheme associated with a singular subalgebroid

Every singular subalgebroid, and in particular every singular foliation $\mathscr{G}$, determines a universal Poisson $C^\infty$-scheme $\mathrm{Spec}(\mathcal{F}(\mathscr{G}))$, which is the analogue of the linear Poisson structure on the dual of a Lie algebroid. We show that the underlying space $X_{\mathcal{F}(\mathscr{G})}$ of $\mathrm{Spec}(\mathcal{F}(\mathscr{G}))$ decomposes as a disjoint union of symplectic manifolds. Furthermore, we extend the Helffer-Nourrigat cone to a singular subalgebroid and show that it is an embedded Poisson $C^\infty$-subscheme of $\mathrm{Spec}(\mathcal{F}(\mathscr{G}))$.

math.SG↗