Equivariant Symplectic Forms

Victor W. Guillemin, Shlomo Sternberg and Jochen Brüning
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Victor W. Guillemin: Massachusetts Institute of Technology, Department of Mathematics
Shlomo Sternberg: Harvard University, Department of Mathematics
Jochen Brüning: Humboldt-Universität Berlin, Institut für Mathematik Mathematisch-Naturwissenschaftliche Fakultät II

Chapter Chapter 9 in Supersymmetry and Equivariant de Rham Theory, 1999, pp 111-147 from Springer

Abstract: Abstract Suppose we are given an action G × M → M. In the Cartan model, an element $$\tilde \omega \in \Omega _G^2(M) = ({\Omega ^2}{(M)^G} \otimes {S^0}({g^*}) \oplus {({\Omega ^0}(M) \otimes {S^1}({g^*}))^G}$$ can be written as $$\tilde \omega = \omega - \phi $$ where ω ∈ Ω2(M) is a two-form invariant under G and $$ \phi \in {({\Omega ^0}(M) \otimes {g^*})^G}$$ can be considered as a G equivariant map, $$\phi :g \to {\Omega ^0}(M) = F(M)$$ from the Lie algebra, g to the space of smooth functions on M. For each ξ ∈ g, ø (ξ) is a smooth function on M, and this function depends linearly on ξ Therefore, for each m ∈ M, the value ø (ξ(m)) depends linearly on, so we can think of ø as defining a map from M to the dual space g* of the Lie algebra of g: $$\phi :M \to {g^*},\left\langle {\phi (m),\xi } \right\rangle : = \phi (\xi )(m).$$

Keywords: Symplectic Form; Symplectic Manifold; Cohomology Ring; Coadjoint Orbit; Minimal Coupling (search for similar items in EconPapers)
Date: 1999
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DOI: 10.1007/978-3-662-03992-2_9

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