# Central extensions of Lie groups preserving a differential form

## TL;DR

This paper generalizes the Kostant-Souriau extension to higher-degree forms on manifolds, constructing central extensions of Lie groups that preserve these forms, using differential characters and transgression techniques.

## Contribution

It introduces a new class of central extensions of Lie groups associated with closed, integral forms of higher degree, extending classical symplectic geometry concepts.

## Key findings

- Constructs central extensions indexed by cohomology groups.
- Shows these extensions integrate to smooth circle group extensions.
- Uses transgression of differential characters in the construction.

## Abstract

Let $M$ be a manifold with a closed, integral $(k+1)$-form $\omega$, and let $G$ be a Fr\'echet-Lie group acting on $(M,\omega)$. As a generalization of the Kostant-Souriau extension for symplectic manifolds, we consider a canonical class of central extensions of $\mathfrak{g}$ by $\mathbb{R}$, indexed by $H^{k-1}(M,\mathbb{R})^*$. We show that the image of $H_{k-1}(M,\mathbb{Z})$ in $H^{k-1}(M,\mathbb{R})^*$ corresponds to a lattice of Lie algebra extensions that integrate to smooth central extensions of $G$ by the circle group $\mathbb{T}$. The idea is to represent a class in $H_{k-1}(M,\mathbb{Z})$ by a weighted submanifold $(S,\beta)$, where $\beta$ is a closed, integral form on $S$. We use transgression of differential characters from $S$ and $ M $ to the mapping space $ C^\infty(S, M) $, and apply the Kostant-Souriau construction on $ C^\infty(S, M) $.

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Source: https://tomesphere.com/paper/1906.03234