Lifting Hamiltonian loops to isotopies in fibrations

TL;DR

This paper explores how Hamiltonian loops in certain geometric structures can be lifted to isotopies in fibrations, providing new insights into the topology of these groups and conditions for loop inequivalence.

Contribution

It establishes a relation between the fundamental group of subgroup of diffeomorphisms and representations of Lie groups, and applies this to Hamiltonian groups of flag and toric varieties.

Findings

Proves a lower bound for the fundamental group of certain diffeomorphism subgroups.

Provides a criterion based on the mass center of Delzant polytopes for loop inequivalence.

Connects the topology of Hamiltonian groups with geometric properties of associated polytopes.

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 ${\rm Diff}(M)$, such that, each vector field $Z\in{\rm Lie}({\mathcal G})$ admits a lift to a preserving connection vector field on ${\mathcal P}$. We prove that $#\,\pi_1({\mathcal G})\geq #\,\Psi(Z(G))$. 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 $\varphi_{\bf b}$ for the the loop in the Hamiltonian group of $M_{\Delta}$ defined by the lattice vector ${\bf b}$. We give a sufficient condition, in terms of the mass center of $\Delta$, for the loops $\varphi_{\bf b}$ and $\varphi_{\bf\tilde b}$ to be homotopically inequivalent.