Quasi-Poisson Manifolds

By A. Alekseev, Y. Kosmann-Schwarzbach, E. Meinrenken. Canadian Journal of Mathematics - Journal Canadien de Mathematiques, 2000

188 citations recorded by the source corpus.

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 ${{\Lambda }^{3}}\mathfrak{g}$ associated to an invariant inner product. We introduce the concept of the fusion of such manifolds, and we relate the quasi-Poisson manifolds to the previously introduced quasi-Hamiltonian manifolds with group-valued moment maps.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8e924ef5290d7070b412f9e9770b1e19afa41567

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. Y. Kosmann-Schwarzbach
2. H. Bursztyn
3. Michel van den Bergh
4. Shahn Majid
5. Dmitry Roytenberg
6. P. Boalch
7. S. E. Akrami
8. G. Massuyeau

In Geometry and Topology

1. Y. Kosmann-Schwarzbach
2. P. Boalch
3. Daisuke Yamakawa
4. Vladimir Turaev
5. G. Massuyeau
6. H. Bursztyn
7. Michel van den Bergh
8. Shahn Majid

In Mathematical Physics

1. Y. Kosmann-Schwarzbach
2. P. Boalch
3. Daisuke Yamakawa
4. Vladimir Turaev
5. G. Massuyeau
6. H. Bursztyn
7. Michel van den Bergh
8. Dmitry Roytenberg

Citations in the corpus

• Deformation quantization of gerbes (2005, background): “he other hand, to quantum groups and their representations. The structures mentioned above (twisted Poisson structures and stacks) play a key role in the geometry of the moduli space, cf. for example [AKM], [AGS], [AMR], [CF], [RS]. There are other motivations for studying deformation quantization o”
• Universal lifting theorem and quasi-Poisson groupoids (2005, background): “29 [1, 2] An Ad-equivariant map J : X → G is called a momentum map for the quasi-Poisson manifold (X,ΠX ) if Π♯X(d(J f)) = (J(Df))X , ∀f ∈ C (G), where D : C∞(G) → C∞(G, g) is defined by Df = 1 2 ∑n i=1( ei + ←− ei )(f)ei.”
• Double Poisson algebras (2004, methodology): “o local systems on P1 whose monodromy lies in the closure of specific conjugacy classes. The result then follows from the work of Atiyah and Bott [3]. Different proofsofthe Atiyah-Bottresultweregivenin [1, 2] using quasi-Hamiltonian reduction and fusion. It is possible to give a proof of Theorem 1.1 i”
• Geometry and braiding of Stokes data; Fission and wild character varieties (2011, background): “m: Proposition 2.8. Suppose M is a smooth affine variety with the structure of quasiHamiltonian G-space. Then the (geometric invariant theory) quotient M/G is a Poisson variety. Proof. It is well-known [3, 2] that the ring of G-invariant functions on M is a Poisson algebra. But by definition the geomet”
• Braided cyclic cocycles and nonassociative geometry (2004, methodology): “Its need is also indicated from Poisson geometry, where the idea of a generalised Poisson bracket violating the usual Jacobi identity is proposed[8].”
• Twisted wild character varieties (2015, background): “Needless to say all the basic properties follow from those established in [3, 2, 1].”
• Quasi-Hamiltonian geometry of meromorphic connections (2002, unknown): “(cid:12)(cid:12) (This sign convention differs from [2] (and agrees with [1]); this leads to sign changes in the quasi-Hamiltonian axioms and the fusion and equivalence theorems.)”
• Courant Algebroids and Poisson Geometry (2008, background): “lting Lagrangian splitting of the Courant algebroid AM = M×ddefines a bi-vector field π∈ X2(M) with 1 2[π,π] = a(Ξ). Up to an irrelevant constant this is the definition of a quasi-Poisson manifold as in [3]. In particular, the group G, and the variety of Lagrangian subalgebras in d= g⊕ gcarry natural q”
• Quasi-Lie Bialgebroids and Twisted Poisson Manifolds (2001, background): “Bivector fields that do not satisfy any integrability condition are not interesting objects unless one imposes an additional structure such as a group action satisfying various types of comapatibility conditions ([7],[1]).”
• Quasi, twisted, and all that... in Poisson geometry and Lie algebroid theory (2003, methodology): “The quasi-Poisson manifolds studied in [2] are manifolds equipped with a bivector, on which such a quasi-Poisson Lie group acts.”
• Quasi-Poisson structures on representation spaces of surfaces (2012, methodology): “dix A we discuss in more detail certain group actions and Lie algebra actions on AN introduced in Section 3. In Appendix B we compare the quasi-Poisson structures on H introduced in this paper and in [AKsM]. Throughout the paper we fix a commutative ring K. Unless stated otherwise, by a module we sha”
• Poisson geometry and Morita equivalence (2004, background): “These Dirac structures are closely related to the theory of quasi-hamiltonian spaces and groupvalued momentum maps [3, 15, 96], as well as to quasi-Poisson manifolds [2, 14].”