[Paper Review] Quasi-Poisson Manifolds

Anton Alekseev, Yvette Kosmann–Schwarzbach+1 more|ArXiv.org|Jun 22, 2000

Homotopy and Cohomology in Algebraic Topology7 references4 citations

TL;DR

This paper introduces quasi-Poisson manifolds—G-manifolds equipped with an invariant bivector field whose Schouten bracket equals the trivector generated by the Cartan 3-tensor associated to an invariant inner product on the Lie algebra. It establishes a deep equivalence between non-degenerate Hamiltonian quasi-Poisson manifolds and quasi-Hamiltonian G-manifolds with group-valued moment maps, enabling the construction of Poisson structures on representation varieties via fusion and reduction.

ABSTRACT

A quasi-Poisson manifold is a G-manifold equipped with an invariant bivector field whose Schouten bracket is the trivector field generated by the invariant element in $\wedge^3 \g$ associated to an invariant inner product. We introduce the concept of the fusion for such manifolds, and we relate quasi-Poisson manifolds to the previously introduced quasi-Hamiltonian manifolds with group-valued moment maps.

Motivation & Objective

• To formalize and study the structure of quasi-Poisson manifolds as G-manifolds with bivector fields satisfying a specific Schouten bracket condition involving the Cartan 3-tensor.
• To develop the theory of fusion and reduction for quasi-Poisson manifolds, generalizing constructions from Poisson geometry.
• To establish a precise correspondence between non-degenerate Hamiltonian quasi-Poisson manifolds and quasi-Hamiltonian G-manifolds with group-valued moment maps.
• To apply this correspondence to construct the standard Poisson structure on the representation variety $\operatorname{Hom}(\pi_1(\Sigma),G)/G$ for surfaces with boundary.
• To introduce and analyze a generalized dynamical r-matrix and prove its solution to a generalized classical dynamical Yang-Baxter equation.

Proposed method

• Define a quasi-Poisson manifold as a G-manifold $M$ with an invariant bivector field $P$ such that $[P,P] = \phi_M$, where $\phi \in \wedge^3\mathfrak{g}$ is the Cartan 3-tensor from an invariant inner product.
• Introduce the concept of fusion of quasi-Poisson manifolds, constructing new structures from products of existing ones.
• Define the moment map $\Phi: M \to G$ for Hamiltonian quasi-Poisson manifolds and show that the triple $(M,P,\Phi)$ satisfies axioms of a quasi-Hamiltonian G-manifold.
• Use the cross-section theorem to prove that every Hamiltonian quasi-Poisson manifold admits a generalized foliation into non-degenerate leaves.
• Construct a generalized dynamical r-matrix $r^{\mathfrak{g}/\mathfrak{h}}$ and prove it satisfies a generalized classical dynamical Yang-Baxter equation via projection and algebraic identities.
• Apply the equivalence between non-degenerate Hamiltonian quasi-Poisson and quasi-Hamiltonian G-manifolds to derive the Poisson structure on $\operatorname{Hom}(\pi_1(\Sigma),G)/G$ via reduction of a fusion product of copies of $G$.

Experimental results

Research questions

• RQ1 How can one generalize Poisson geometry to include group-valued moment maps via a new class of G-manifolds?
• RQ2 What is the precise relationship between Hamiltonian quasi-Poisson manifolds and quasi-Hamiltonian G-manifolds with group-valued moment maps?
• RQ3 Can the Poisson structure on representation varieties $\operatorname{Hom}(\pi_1(\Sigma),G)/G$ be derived from a geometric reduction procedure?
• RQ4 What is the role of the Cartan 3-tensor $\phi \in \wedge^3\mathfrak{g}$ in defining the quasi-Poisson condition $[P,P] = \phi_M$?
• RQ5 How does the generalized dynamical r-matrix $r^{\mathfrak{g}/\mathfrak{h}}$ satisfy a generalized classical dynamical Yang-Baxter equation?

Key findings

• Every Hamiltonian quasi-Poisson manifold admits a generalized foliation into non-degenerate Hamiltonian quasi-Poisson submanifolds.
• Every non-degenerate Hamiltonian quasi-Poisson manifold carries an invariant 2-form $\omega$ such that $(M,\omega,\Phi)$ satisfies the axioms of a quasi-Hamiltonian G-manifold with group-valued moment map.
• Conversely, every quasi-Hamiltonian G-manifold with group-valued moment map carries a non-degenerate quasi-Poisson structure with the same moment map.
• The Poisson structure on the representation variety $\operatorname{Hom}(\pi_1(\Sigma),G)/G$ for an oriented surface $\Sigma$ with boundary arises via reduction of a fusion product of copies of $G$ equipped with quasi-Poisson structures.
• The generalized dynamical r-matrix $r^{\mathfrak{g}/\mathfrak{h}}$ satisfies the generalized classical dynamical Yang-Baxter equation, as proven via projection and algebraic identities in the Lie algebra.
• The quasi-Poisson bivector on $G$ is related to the Poisson bivector on the dual of the Lie algebra of the central extension of the loop group $LG$, via a limit process involving cotangent functions and residue sums.