Timeline for line bundle on affine grassmannian and central extension

4 events

when toggle format	what		by	license	comment
Oct 23, 2015 at 0:51	comment	added	Allen Knutson		On a symplectic manifold $(M,\omega)$, the map $Fun(M) \to Vec(M)$ taking $f\mapsto \omega^{-1}(df)$ is a Lie algebra homorphism and central extension of its image. Pull back that central extension to $L\mathfrak k$. If it were trivial, then $L\mathfrak k$ would factor through $Fun(\Omega K)$, giving dually an $LK$-equivariant map $\Omega K\to (L\mathfrak k)^*$. But none exists. This general idea (not this example) is in Guillemin-Sternberg "Symplectic Techniques in Physics".
Oct 22, 2015 at 16:04	comment	added	prochet		Why the fact that $\Omega K$ is not a coadjoint orbit of $LK$ implies that the extension is non-trivial.
Oct 22, 2015 at 12:35	comment	added	Allen Knutson		In the more topological version, with $K$ a maximal compact group of $G$, $Gr$ sits densely inside the symplectic manifold $\Omega K$. The group $LK$ acts symplectically and transitively on it, but $\Omega K$ is not a coadjoint orbit of $LK$, only of this central extension -- hence it's nontrivial. Lots of this is in Pressley and Segal's Loop Groups.
Oct 22, 2015 at 2:13	history	asked	prochet	CC BY-SA 3.0