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- W68058592 abstract "Let M be a differentiable manifold, (L;mathop to limits^p ;M) a complex line bundle and ∇ a linear connection on L. Its curvature form gives rise to a closed 2-form on M. Conversely, ω ∈ Z 2(M) arises in the said fashion, if it is of an integral de Rham class. Below we assume, that ω is also real and nondegenerate. We consider the totality e(M) of all smooth functions on M as a Lie algebra with respect to the Poisson bracket { , }. Following B. Kostant (cf.[2]), we can canonically associate with the said data a morphism δ of e(M), called the prequantization map, in the Lie algebra of endomorphisms of all smooth sections e(L) by setting $$delta (varphi ) = {X_varphi } - 2pi ivarphi ;{rm{ (}}varphi in varepsilon ({rm M}){rm{)}}$$where the Hamiltonian vector field X ϕ is defined by dϕ = −ι(X ϕ)ω. — Given an integrable Lagrangean distribution H on M, we write VH(M) for the Lie algebra of vector fields with values in H. We call a section polarized, if it is annihilated by all ∇ X (X ∈ V H (M)), and shall write e0(L) for their totality. Let e0 be the collection of all elements of e(M), which satisfy Xϕ ≡ 0 (ϕ ∈ e(M), X ∈ V H (M)). e0 is a maximal abelian subalgebra of e(M); we denote its normalizer by e1. Then the image of e1 via δ leaves e0(L) invariant. The restriction of δ to e1 and e0(L) is the quantization map." @default.
- W68058592 created "2016-06-24" @default.
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- W68058592 date "1990-01-01" @default.
- W68058592 modified "2023-09-24" @default.
- W68058592 title "On a Property of the Quantization Map for the Coadjoint Orbits of Connected Lie Groups" @default.
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- W68058592 doi "https://doi.org/10.1007/978-1-4612-4486-8_9" @default.
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