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Kostant-Souriau extension

Contents

Idea

Traditional notion

Given a presymplectic manifold (X,Ο‰)(X,\omega) there is the Poisson bracket Lie algebra 𝔓𝔬𝔦𝔰𝔰𝔬𝔫(X.Ο‰)\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 ℝ\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.

Refinement to higher geometry

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

These in turn are special cases of L-∞ algebra extension/∞-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:

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

(Ω𝔾)FlatConn(X)β†’QuantMorph(X,βˆ‡)β†’HamSympl(X,βˆ‡) (\Omega \mathbb{G})\mathbf{FlatConn}(X) \to \mathbf{QuantMorph}(X,\nabla) \to \mathbf{HamSympl}(X,\nabla)
nngeometrystructureunextended structureextension byquantum extension
∞\inftyhigher prequantum geometrycohesive ∞-groupHamiltonian symplectomorphism ∞-groupmoduli ∞-stack of (Ω𝔾)(\Omega \mathbb{G})-flat ∞-connections on XXquantomorphism ∞-group
1symplectic geometryLie algebraHamiltonian vector fieldsreal numbersHamiltonians under Poisson bracket
1Lie groupHamiltonian symplectomorphism groupcircle groupquantomorphism group
22-plectic geometryLie 2-algebraHamiltonian vector fieldsline Lie 2-algebraPoisson Lie 2-algebra
2Lie 2-groupHamiltonian 2-plectomorphismscircle 2-groupquantomorphism 2-group
nnn-plectic geometryLie n-algebraHamiltonian vector fieldsline Lie n-algebraPoisson Lie n-algebra
nnsmooth n-groupHamiltonian n-plectomorphismscircle n-groupquantomorphism n-group

(extension are listed for sufficiently connected XX)

References

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 ∞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.