Kostant-Souriau extension

Given a presymplectic manifold $(X,\omega)$ there is the Poisson bracket Lie algebra $\mathfrak{Poisson}(X.\omega)$ of Hamiltonians and their Hamiltonian vector fields. This is a extension of Lie algebras of the Lie algebra of just Hamiltonian vecotor fields. Over a connected manifold it is an extension by $\mathbb{R}$. The Lie integration of this extension is the quantomorphism group-extension of the group of Hamiltonian symplectomorphisms.

These extensions are called **Kostant-Souriau extensions** after Bertram Kostant and Jean-Marie Souriau. They play a central role in geometric quantization.

For $(X,\omega)$ a pre-n-plectic manifold there is an L-β algebra extension that generalizes the Kostant Souriau extension: the Poisson bracket L-β algebra (Rogers).

These in turn are special cases of L-β algebra extension/β-group extensions which are variants of the higher Atiyah groupoid-extensions that exist in general in cohesive higher geometry (Hgpt), as indicated in the following table:

**higher and integrated Kostant-Souriau extensions**:

(β-group extension of β-group of bisections of higher Atiyah groupoid for $\mathbb{G}$-principal β-connection)

$(\Omega \mathbb{G})\mathbf{FlatConn}(X) \to \mathbf{QuantMorph}(X,\nabla) \to \mathbf{HamSympl}(X,\nabla)$

(extension are listed for sufficiently connected $X$)

The traditional Kostant-Souriau extension appears for instance prop. 2.3.9 in

- Jean-Luc Brylinski,
*Loop spaces, characteristic classes and geometric quantization*, BirkhΓ€user (1993)

The refinement to n-plectic geometry is due to

- Chris Rogers,
*$L_\infty$ algebras from multisymplectic geometry*,Letters in Mathematical Physics April 2012, Volume 100, Issue 1, pp 29-50 (arXiv:1005.2230, journal).

A general characterization in higher geometry is in

Last revised on August 16, 2018 at 18:52:03. See the history of this page for a list of all contributions to it.