Lifting Hamiltonian loops to isotopies in fibrations

Abstract

Let $G$ be a Lie group, $H$ a closed subgroup and $M$ the homogeneous space $G/H$. Each representation $\Psi$ of $H$ determines a $G$-equivariant principal bundle $\mathcal{P}$ on $M$ endowed with a $G$-invariant connection. We consider subgroups $\mathcal{G}$ of the diffeomorphism group $\mathrm{Diff}(M)$, such that, each vector field $Z\in \mathrm{Lie}(\mathcal{G})$ admits a lift to a preserving connection vector field on $\mathcal{P}$. We prove that $#\,π_1({\mathcal G})\geq #\,Ψ(Z(G))$. This relation is applicable to subgroups $\mathcal{G}$ of the Hamiltonian groups of the flag varieties of a semisimple group $G$. Let $M_{\Delta }$ be the toric manifold determined by the Delzant polytope $\Delta$. We put ${\phi }_{\mathbf{b}}$ for the the loop in the Hamiltonian group of $M_{\Delta }$ defined by the lattice vector $\mathbf{b}$. We give a sufficient condition, in terms of the mass center of $\Delta$, for the loops ${\phi }_{\mathbf{b}}$ and ${\phi }_{\tilde{\mathbf{b}}}$ to be homotopically inequivalent.

23 pages, 1 figure. To be published in Int. J. Geom. Methods Mod. Physics

References in corpus (1)

• Continuous families of Hamiltonian torus actions