Abelian extensions via prequantization

TL;DR

This paper extends the concept of prequantization central extensions to abelian extensions of diffeomorphism groups preserving vector-valued 2-forms, unifying various extensions through pull-back constructions.

Contribution

It introduces a generalized framework for abelian extensions of diffeomorphism groups preserving vector-valued 2-forms, broadening the scope of prequantization methods.

Findings

Every abelian extension of a simply connected Lie group can be realized as a pull-back of the constructed extension.

Generalization from scalar to vector-valued 2-forms in the context of diffeomorphism groups.

Establishment of a unifying approach for abelian extensions via prequantization.

Abstract

We generalize the prequantization central extension of a group of diffeomorphisms preserving a closed 2-form \omega (\omega-invariant diffeomorphisms) to an abelian extension of a group of diffeomorphisms preserving a closed vector valued 2-form \omega, up to a linear isomorphism (\omega-equivariant diffeomorphisms). Every abelian extension of a simply connected Lie group can be obtained as the pull-back of such a prequantization abelian extension.