The hyperbolic moduli space of flat connections and the isomorphism of symplectic multiplicity spaces

1996/03/29 by Anton Alekseev, Anton Yu. Alekseev, Alekseev, Anton Yu. +3
Mathematics · #Advanced Algebra and Geometry #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #dg-ga #math.DG #math.QA #q-alg

paper · pdf · doi:10.48550/arxiv.dg-ga/9603017

19 pages, AMS LaTeX

arxiv created 1996/03/29 · openalex publication_date 1996/03/29 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G be a simple complex Lie group, \algg be its Lie algebra, K be a maximal compact form of G and \algk be a Lie algebra of K. We denote by X→ X the anti-involution of \algg which singles out the compact form \algk. Consider the space of flat \algg-valued connections on a Riemann sphere with three holes which satisfy the additional condition A(z)=-A(z). We call the quotient of this space over the action of the gauge group g(z)=g^-1(z) a hyperbolic moduli space of flat connections. We prove that the following three symplectic spaces are isomorphic: 1. The hyperbolic moduli space of flat connections. 2. The symplectic multiplicity space obtained as symplectic quotient of the triple product of co-adjoint orbits of K. 3. The Poisson-Lie multiplicity space equal to the Poisson quotient of the triple product of dressing orbits of K.

Citations

• On Poisson actions of compact Lie groups on symplectic manifolds

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