Integrability of central extensions of the Poisson Lie algebra via prequantization

TL;DR

This paper constructs central S^1-extensions of the quantomorphism group and Hamiltonian diffeomorphisms on symplectic manifolds, linking geometric prequantization with algebraic cocycles and Lie group extensions.

Contribution

It provides a geometric method to realize central extensions of the Poisson Lie algebra and explicitly describes the integrable cocycles, advancing the understanding of symplectic and contact transformation groups.

Findings

Explicit description of lattice of integrable cocycles

Construction of nontrivial central S^1-extensions

Extensions of Lie groups acting by contact transformations

Abstract

We present a geometric construction of central S^1-extensions of the quantomorphism group of a prequantizable, compact, symplectic manifold, and explicitly describe the corresponding lattice of integrable cocycles on the Poisson Lie algebra. We use this to find nontrivial central S^1-extensions of the universal cover of the group of Hamiltonian diffeomorphisms. In the process, we obtain central S^1-extensions of Lie groups that act by exact strict contact transformations.