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| In the study of [[mathematics]] and especially [[differential geometry]], '''fundamental vector fields''' are an instrument that describes the infinitesimal behaviour of a [[Smooth function|smooth]] [[Lie group]] action on a [[Differentiable manifold#Definition|smooth manifold]]. Such [[vector field#Vector fields on manifolds|vector fields]] find important applications in the study of [[Lie theory]], [[symplectic geometry]], and the study of [[Moment map#Hamiltonian group actions|Hamiltonian group actions]].
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| ==Motivation==
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| Important to applications in mathematics and [[physics]]<ref name="HouBook">{{Citation | last1=Hou | first1=Bo-Yu | title=Differential Geometry for Physicists | publisher=[[World Scientific|World Scientific Publishing Company]] | isbn=978-9810231057 | year=1997}}</ref> is the notion of a [[Flow (mathematics)|flow]] on a manifold. In particular, if <math> M </math> is a smooth manifold and <math> X</math> is a smooth [[Vector field#Vector fields on manifolds|vector field]], one is interested in finding [[integral curve]]s to <math> X </math>. More precisely, given <math> p \in M </math> one is interested in curves <math> \gamma_p: \mathbb R \to M </math> such that
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| :<math> \gamma_p'(t) = X_{\gamma_p(t)}, \qquad \gamma_p(0) = p, </math>
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| for which local solutions are guaranteed by the [[Ordinary differential equation#Existence and uniqueness of solutions|Existence and Uniqueness Theorem of Ordinary Differential Equations]]. If <math> X </math> is furthermore a [[Vector field#Complete vector fields|complete vector field]], then the flow of <math> X </math>, defined as the collection of all integral curves for <math> X </math>, is a [[diffeomorphism]] of <math> M</math>. The flow <math> \phi_X: \mathbb R \times M \to M </math> given by <math> \phi_X(t,p) = \gamma_p(t) </math> is in fact an [[group action|action]] of the additive Lie group <math> (\mathbb R,+) </math> on <math> M</math>.
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| Conversely, every smooth action <math> A:\mathbb R \times M \to M </math> defines a complete vector field <math> X </math> via the equation
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| : <math> X_p = \left.\frac{d}{dt}\right|_{t=0} A(t,p). </math>
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| It is then a simple result<ref name="da Silva">{{Cite book | last1=Canas da Silva | first1=Ana | author1-link=Ana Canas da Silva | title=Lectures on Symplectic Geometry | publisher=Springer | isbn=978-3540421955| year=2008}}</ref> that there is a bijective correspondence between <math> \mathbb R </math> actions on <math> M </math> and complete vector fields on <math> M </math>.
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| In the language of flow theory, the vector field <math> X </math> is called the ''infinitesimal generator''.<ref name="Lee">{{Cite book | last1 = Lee | first1 = John | title=Introduction to Smooth Manifolds | publisher=Springer | isbn= 0-387-95448-1 | year=2003}}</ref> Intuitively, the behaviour of the flow at each point corresponds to the "direction" indicated by the vector field. It is a natural question to ask whether one may establish a similar correspondence between vector fields and more arbitrary Lie group actions on <math> M </math>
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| ==Definition==
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| Let <math> G </math> be a Lie group with corresponding [[Lie algebra#Relation to Lie groups|Lie algebra]] <math> \mathfrak g </math>. Furthermore, let <math> M </math> be a smooth manifold endowed with a [[Lie group action|smooth action]] <math> A : G \times M \to M </math>. Denote the map <math> A_p: G \to M </math> such that <math> A_p(g) = A(g,p) </math>, called the ''orbit map of <math> A</math> corresponding to <math> p </math>''.<ref name="Audin">{{Cite book | last1 = Audin | first1 = Michèle | title=Torus Actions on Symplectic manifolds | publisher=Birkhäuser | isbn= 3-7643-2176-8 | year=2004}}</ref> For <math> X \in \mathfrak g </math>, the fundamental vector field <math> X^\# </math> corresponding to <math> X </math> is any of the following equivalent definitions:<ref name="da Silva"/><ref name="Audin"/><ref name="Libermann">{{Cite book | last1 = Libermann | first1 = Paulette | last2 = Marle | first2 = Charles-Michel | title=Symplectic Geometry and Analytical Mechanics | publisher=Springer | isbn= 978-9027724380 | year=1987}}</ref>
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| *<math> X^\#_p = d_e A_p(X) </math>
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| *<math> X^\#_p = d_{(e,p)}A\left(X,0_{T_p M}\right) </math>
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| *<math> X^\#_p = \left. \frac{d}{dt} \right|_{t=0} A\left( \exp(tX), p \right)</math>
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| where <math> d </math> is the [[differential of a smooth map]] and <math> 0_{T_pM} </math> is the [[null vector|zero vector]] in the [[vector space]] <math> T_p M</math>.
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| The map <math> \mathfrak g \to \Gamma(TM), X \mapsto X^\# </math> can then be shown to be a [[Lie algebra#Homomorphisms, subalgebras, and ideals|Lie algebra homomorphism]].<ref name="Libermann"/>
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| ==Applications==
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| ===Lie groups===
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| The Lie algebra of a Lie group <math> G </math> may be identified with either the left- or right-invariant vector fields on <math> G </math>. It is a well known result<ref name="Lee"/> that such vector fields are isomorphic to <math> T_e G </math>, the tangent space at identity. In fact, if we let <math> G </math> act on itself via right-multiplication, the corresponding fundamental vector fields are precisely the left-invariant vector fields.
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| ===Hamiltonian group actions===
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| In the [[Fundamental vector field#Motivation|motivation]], it was shown that there is a bijective correspondence between smooth <math> \mathbb R </math> actions and complete vector fields. Similarly, there is a bijective correspondence between symplectic actions (the induced [[diffeomorphisms]] are all [[symplectomorphisms]]) and complete [[symplectic vector field]]s.
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| A closely related idea is that of [[Hamiltonian vector field]]s. Given a symplectic manifold <math> (M,\omega) </math>, we say that <math> X_H</math> is a Hamiltonian vector field if there exists a [[smooth function]] <math> H: M \to \mathbb R </math> satisfying
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| :<math> dH = \iota_{X_H}\omega </math>
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| where the map <math> \iota </math> is the [[interior product]]. This motivatives the definition of a ''Hamiltonian group action'' as follows: If <math> G </math> is a Lie group with Lie algebra <math> \mathfrak g </math> and <math> A: G\times M \to M </math> is a group action of <math> G </math> on a smooth manifold <math> M </math>, then we say that <math> A </math> is a Hamiltonian group action if there exists a [[moment map]] <math> \mu: M \to \mathfrak g^* </math> such that for each <math> X \in \mathfrak g </math>,
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| : <math> d\mu^X = \iota_{X^\#}\omega, </math>
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| where <math> \mu^X:M \to \mathbb R, p \mapsto \langle \mu(p),X \rangle </math> and <math> X^\# </math> is the fundamental vector field of <math> X </math>
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| ==References==
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| <references />
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| [[Category:Lie groups]]
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| [[Category:Symplectic geometry]]
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| [[Category:Hamiltonian mechanics]]
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| [[Category:Smooth manifolds]]
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