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Let G be a Lie group and g its Lie algebra. We develop a theory of quasiPoisson structures relative to a not necessarily non-degenerate Ad-invariantsymmetric 2-tensor in the tensor square of g and one of general not necessarilynon-degenerate quasi Hamiltonian structures relative to a not necessarilynon-degenerate Ad-invariant symmetric bilinear form on g, a quasi Poissonstructure being given by a skew bracket of two variables such that suitabledata defined in terms of G as symmetry group involving the 2-tensor measure howthat bracket fails to satisfy the Jacobi identity. The present approachinvolves a novel concept of momentum mapping and yields, in the non-degeneratecase, a bijective correspondence between non-degenerate quasi Poissonstructures and non-degenerate quasi Hamiltonian structures. The new theoryapplies to various not necessarily non-singular moduli spaces and yieldsthereupon, via reduction with respect to an appropriately defined momentummapping, not necessarily non-degenerate ordinary Poisson structures. Amongthese moduli spaces are representation spaces, possibly twisted, of thefundamental group of a Riemann surface, possibly punctured, and moduli spacesof semistable holomorphic vector bundles as well as Higgs bundle moduli spaces.In the non-degenerate case, such a Poisson structure comes down to a stratifiedsymplectic one of the kind explored in the literature and recovers, e.g., thesymplectic part of a K\"ahler structure introduced by Narasimhan and Seshadrifor moduli spaces of stable holomorphic vector bundles on a curve. In thealgebraic setting, these moduli spaces arise as not necessarily non-singularaffine not necessarily non-degenerate Poisson varieties. A side result is anexplicit equivalence between extended moduli spaces and quasi Hamiltonianspaces independently of gauge theory.

Quasi Poisson structures, weakly quasi Hamiltonian structures, and Poisson geometry of various moduli spaces