Quasi-Poisson Manifolds

Alekseev, Anton; Kosmann–Schwarzbach, Yvette; Meinrenken, Eckhard

doi:10.4153/cjm-2002-001-5

Cited by 158 publications

(420 citation statements)

References 10 publications

(36 reference statements)

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“…In order to obtain symplectic structures on the orbit space Hom(π g,l , U )/U of this action (that is, the space of equivalence classes of representations), one has to fix the conjugacy classes of the generators c 1 , ... , c l representing homotopy classes of loops around the punctures s 1 , ..., s l (see for instance [6]). Otherwise, one only obtains a Poisson structure (see for instance [2]). Observe that the conjugacy classes of the a i , b i need not be fixed.…”

Section: Introductionmentioning

confidence: 99%

Decomposable representations and Lagrangian submanifolds of moduli spaces associated to surface groups

Schaffhauser

2008

Math. Ann.

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Abstract. The importance of explicit examples of Lagrangian submanifolds of moduli spaces is revealed by papers such as [9,25]: given a 3-manifold M with boundary ∂M = Σ, Dostoglou and Salamon use such examples to obtain a proof of the Atiyah-Floer conjecture relating the symplectic Floer homology of the representation space Hom(π 1 (Σ = ∂M ), U )/U (associated to an explicit pair of Lagrangian submanifolds of this representation space) and the instanton homology of the 3-manifold M . In the present paper, we construct a Lagrangian submanifold of the space of representations M g,l := Hom C (π g,l , U )/U of the fundamental group π g,l of a punctured Riemann surface Σ g,l into an arbitrary compact connected Lie group U . This Lagrangian submanifold is obtained as the fixed-point set of an anti-symplectic involutionβ defined on M g,l . We show that the involutionβ is induced by a form-reversing involution β defined on the quasi-Hamiltonian space (The fact thatβ has a non-empty fixed-point set is a consequence of the real convexity theorem for group-valued momentum maps proved in [28]. The notion of decomposable representation provides a geometric interpretation of the Lagrangian submanifold thus obtained.

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Section: Introductionmentioning

confidence: 99%

Decomposable representations and Lagrangian submanifolds of moduli spaces associated to surface groups

Schaffhauser

2008

Math. Ann.

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“…We see that C q (M) is not the quantization of a usual Poisson manifold but of a 'quasi-Poisson' manifold. Such a weaker concept was proposed recently in [8] and we see that we have succeed in quantizing it using cotwisting. We see, moreover, that the quantum quasispace C q (M) remains covariant but under the quantum group C q (G) (viewed as a cotriangular coquasiHopf algebra).…”

Section: Quantum Quasimanifolds Covariant Under Quantum Groupsmentioning

confidence: 99%

“…Indeed, the need for some kind of nonassociative geometry is hinted at from several directions in mathematical physics including string theory. Its need is also indicated from Poisson geometry, where the idea of a generalised Poisson bracket violating the usual Jacobi identity is proposed [8]. It turns out that an adequate class that appears to cover such examples is based on the use of Drinfeld type cotwists, but not for (coquasi)Hopf algebras H as above.…”

Section: Introductionmentioning

confidence: 99%

“…Section 5.3 also outlines the theory applied to formal deformation theory, where we obtain the quasialgebras C q (G) as mentioned above, using Drinfeld's associator obtained from solving the Knizhnik-Zamalochikov equations. We can also in principle use cotwising to deformation-quantise the quasiPoisson manifold structure on a G-manifold M proposed in [8] (which was not achieved before), which we do as a quasialgebra C q [M]. We also construct the differential calculus and braided cyclic cocycles on all these quasialgebras.…”

Section: Introductionmentioning

confidence: 99%

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Braided cyclic cocycles and nonassociative geometry

Akrami

Majid

2004

Journal of Mathematical Physics

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We use monoidal category methods to study the noncommutative geometry of nonassociative algebras obtained by a Drinfeld-type cochain twist. These are the so-called quasialgebras and include the octonions as braided-commutative but nonassociative coordinate rings, as well as quasialgebra versions C q (G) of the standard q-deformation quantum groups. We introduce the notion of ribbon algebras in the category, which are algebras equipped with a suitable generalised automorphism σ, and obtain the required generalisation of cyclic cohomology. We show that this braided cyclic cocohomology is invariant under a cochain twist. We also extend to our generalisation the relation between cyclic cohomology and differential calculus on the ribbon quasialgebra. The paper includes differential calculus and cyclic cocycles on the octonions as a finite nonassociative geometry, as well as the algebraic noncommutative torus as an associative example.

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“…The modular classes of Leibniz algebroids [58,123], of Nambu-Poisson structures [57,56], of Jacobi manifolds [121] and Jacobi algebroids (also called generalized Lie algebroids) [60] have been defined, their unimodularity implying a homology/cohomology duality, as well as those of symplectic supermanifolds [102]. The modular class of quasi-Poisson G-manifolds was defined by Alekseev et al in [6].…”

Section: Appendix: Additional References and Conclusionmentioning

confidence: 99%