Quasi Poisson structures, weakly quasi ...
Document type :
Article dans une revue scientifique: Article original
Permalink :
Title :
Quasi Poisson structures, weakly quasi Hamiltonian structures, and Poisson geometry of various moduli spaces
Author(s) :
Journal title :
Journal of Geometry and Physics
Pages :
104851
Publisher :
Elsevier
Publication date :
2023-08
ISSN :
0393-0440
English keyword(s) :
Quasi Poisson structure
weakly quasi Hamiltonian structure
extended moduli space
momentum mapping in quasi setting
representation space
analytic representation variety
algebraic representation variety
analytic quasi Hamiltonian reduction
algebraic quasi Hamiltonian reduction
analytic quasi Poisson reduction
algebraic quasi Poisson reduction
affine Poisson variety
Dirac structure
weakly quasi Hamiltonian structure
extended moduli space
momentum mapping in quasi setting
representation space
analytic representation variety
algebraic representation variety
analytic quasi Hamiltonian reduction
algebraic quasi Hamiltonian reduction
analytic quasi Poisson reduction
algebraic quasi Poisson reduction
affine Poisson variety
Dirac structure
HAL domain(s) :
Mathématiques [math]
English abstract : [en]
Let G be a Lie group and g its Lie algebra. We develop a theory of quasi Poisson structures relative to a not necessarily non-degenerate Ad-invariant symmetric 2-tensor in the tensor square of g and one of general not ...
Show more >Let G be a Lie group and g its Lie algebra. We develop a theory of quasi Poisson structures relative to a not necessarily non-degenerate Ad-invariant symmetric 2-tensor in the tensor square of g and one of general not necessarily non-degenerate quasi Hamiltonian structures relative to a not necessarily non-degenerate Ad-invariant symmetric bilinear form on g, a quasi Poisson structure being given by a skew bracket of two variables such that suitable data defined in terms of G as symmetry group involving the 2-tensor measure how that bracket fails to satisfy the Jacobi identity. The present approach involves a novel concept of momentum mapping and yields, in the non-degenerate case, a bijective correspondence between non-degenerate quasi Poisson structures and non-degenerate quasi Hamiltonian structures. The new theory applies to various not necessarily non-singular moduli spaces and yields thereupon, via reduction with respect to an appropriately defined momentum mapping, not necessarily non-degenerate ordinary Poisson structures. Among these moduli spaces are representation spaces, possibly twisted, of the fundamental group of a Riemann surface, possibly punctured, and moduli spaces of semistable holomorphic vector bundles as well as Higgs bundle moduli spaces. In the non-degenerate case, such a Poisson structure comes down to a stratified symplectic one of the kind explored in the literature and recovers, e.g., the symplectic part of a Kähler structure introduced by Narasimhan and Seshadri for moduli spaces of stable holomorphic vector bundles on a curve. In the algebraic setting, these moduli spaces arise as not necessarily non-singular affine not necessarily non-degenerate Poisson varieties. A side result is an explicit equivalence between extended moduli spaces and quasi Hamiltonian spaces independently of gauge theory.Show less >
Show more >Let G be a Lie group and g its Lie algebra. We develop a theory of quasi Poisson structures relative to a not necessarily non-degenerate Ad-invariant symmetric 2-tensor in the tensor square of g and one of general not necessarily non-degenerate quasi Hamiltonian structures relative to a not necessarily non-degenerate Ad-invariant symmetric bilinear form on g, a quasi Poisson structure being given by a skew bracket of two variables such that suitable data defined in terms of G as symmetry group involving the 2-tensor measure how that bracket fails to satisfy the Jacobi identity. The present approach involves a novel concept of momentum mapping and yields, in the non-degenerate case, a bijective correspondence between non-degenerate quasi Poisson structures and non-degenerate quasi Hamiltonian structures. The new theory applies to various not necessarily non-singular moduli spaces and yields thereupon, via reduction with respect to an appropriately defined momentum mapping, not necessarily non-degenerate ordinary Poisson structures. Among these moduli spaces are representation spaces, possibly twisted, of the fundamental group of a Riemann surface, possibly punctured, and moduli spaces of semistable holomorphic vector bundles as well as Higgs bundle moduli spaces. In the non-degenerate case, such a Poisson structure comes down to a stratified symplectic one of the kind explored in the literature and recovers, e.g., the symplectic part of a Kähler structure introduced by Narasimhan and Seshadri for moduli spaces of stable holomorphic vector bundles on a curve. In the algebraic setting, these moduli spaces arise as not necessarily non-singular affine not necessarily non-degenerate Poisson varieties. A side result is an explicit equivalence between extended moduli spaces and quasi Hamiltonian spaces independently of gauge theory.Show less >
Language :
Anglais
Peer reviewed article :
Oui
Audience :
Internationale
Popular science :
Non
Collections :
Source :
Submission date :
2025-01-24T14:23:25Z
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