KN
Karl-Hermann Neeb
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1
Using a nonlinear version of the tautological bundle over Graßmannians, we construct a transgression map for differential characters from M to the nonlinear Graßmannian GrS(M) of submanifolds of M of a fixed type S. In particular, we obtain prequantum circle bundles of the nonlinear Graßmannian endowed with the Marsden–Weinstein symplectic form. The associated Kostant–Souriau prequantum extension yields central Lie group extensions of a group of volume-preserving diffeomorphisms integrating Lichnerowicz cocycles.
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Using a nonlinear version of the tautological bundle over Graßmannians, we construct a transgression map for differential characters from M to the nonlinear Graßmannian GrS(M) of submanifolds of M of a fixed type S. In particular, we obtain prequantum circle bundles of the nonlinear Graßmannian endowed with the Marsden–Weinstein symplectic form. The associated Kostant–Souriau prequantum extension yields central Lie group extensions of a group of volume-preserving diffeomorphisms integrating Lichnerowicz cocycles.
Let M be a manifold with a closed, integral (k+1)-form ω, and let G be a Fréchet–Lie group acting on (M,ω). As a generalization of the Kostant–Souriau extension for symplectic manifolds, we consider a canonical class of central extensions of g by R, indexed by Hk−1(M,R)∗. We show that the image of Hk−1(M,Z) in Hk−1(M,R)∗ corresponds to a lattice of Lie algebra extensions that integrate to smooth central extensions of G by the circle group T. The idea is to represent a class in Hk−1(M,Z) by a weighted submanifold (S,β), where β is a closed, integral form on S. We use transgression of differential characters from S and M to the mapping space C∞(S,M) and apply the Kostant–Souriau construction on C∞(S,M).
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Let M be a manifold with a closed, integral (k+1)-form ω, and let G be a Fréchet–Lie group acting on (M,ω). As a generalization of the Kostant–Souriau extension for symplectic manifolds, we consider a canonical class of central extensions of g by R, indexed by Hk−1(M,R)∗. We show that the image of Hk−1(M,Z) in Hk−1(M,R)∗ corresponds to a lattice of Lie algebra extensions that integrate to smooth central extensions of G by the circle group T. The idea is to represent a class in Hk−1(M,Z) by a weighted submanifold (S,β), where β is a closed, integral form on S. We use transgression of differential characters from S and M to the mapping space C∞(S,M) and apply the Kostant–Souriau construction on C∞(S,M).
For an infinite dimensional Lie group G modelled on a locally convex Lie algebra g, we prove that every smooth projective unitary representation of G corresponds to a smooth linear unitary representation of a Lie group extension G♯ of G. (The main point is the smooth structure on G♯.) For infinite dimensional Lie groups G which are 1-connected, regular, and modelled on a barrelled Lie algebra g, we characterize the unitary g-representations which integrate to G. Combining these results, we give a precise formulation of the correspondence between smooth projective unitary representations of G, smooth linear unitary representations of G♯, and the appropriate unitary representations of its Lie algebra g♯.
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For an infinite dimensional Lie group G modelled on a locally convex Lie algebra g, we prove that every smooth projective unitary representation of G corresponds to a smooth linear unitary representation of a Lie group extension G♯ of G. (The main point is the smooth structure on G♯.) For infinite dimensional Lie groups G which are 1-connected, regular, and modelled on a barrelled Lie algebra g, we characterize the unitary g-representations which integrate to G. Combining these results, we give a precise formulation of the correspondence between smooth projective unitary representations of G, smooth linear unitary representations of G♯, and the appropriate unitary representations of its Lie algebra g♯.
Motivated by positive energy representations , we classify those continuous central extensions of the compactly supported gauge Lie algebra that are covariant under a 1-parameter group of transformations of the base manifold.
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Motivated by positive energy representations , we classify those continuous central extensions of the compactly supported gauge Lie algebra that are covariant under a 1-parameter group of transformations of the base manifold.