Timeline for Exact DG Poisson algebra
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Oct 8, 2012 at 17:32 | history | edited | DamienC | CC BY-SA 3.0 |
Quasi-iso is an iso
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| Oct 8, 2012 at 17:31 | comment | added | DamienC | So, the quasi-isomorphism is actually an isomorphism of dg algebras. | |
| Oct 8, 2012 at 17:28 | comment | added | DamienC | @igor khavkine: there is an isomorphism $TX\to T^*X$ that sends a vector $v$ to the one-form $\omega(v,-)$. It extends to an algebra morphism from polyvectors to forms (wedge product on both sides). You can chrck in local coordinates that $[\pi,-]$ is sent to $d$. | |
| Oct 8, 2012 at 17:27 | comment | added | Nicola Ciccoli | The fact that Poisson and De Rham cohomology are isomorphic on symplectic manifolds is not difficlt. The Poisson bivector defines a "sharp map" from k-forms to k-multivector fields, just by extending the map between 1-forms and 1-vector fields given by contractipn with a 2-bivector. This map intertwines coboundaries and intertwines cup products. It is an isomorphism already at the chain level, being invertible. Every book on Poisson manifold has this remarked; maybe the easiest source (depends on tastes) is Vaisman "lectures on the geometry of Poisson manifolds" birkhauser. | |
| Oct 8, 2012 at 12:19 | comment | added | Igor Khavkine | Do you have a reference or simple argument for the quasi-isomorphism? | |
| Oct 8, 2012 at 7:51 | history | answered | DamienC | CC BY-SA 3.0 |