Fundamental vector field

In the study of mathematics and especially differential geometry, fundamental vector fields are an instrument that describes the infinitesimal behaviour of a smooth Lie group action on a smooth manifold. Such vector fields find important applications in the study of Lie theory, symplectic geometry, and the study of Hamiltonian group actions.

Motivation

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Important to applications in mathematics and physics[1] is the notion of a flow on a manifold. In particular, if   is a smooth manifold and   is a smooth vector field, one is interested in finding integral curves to  . More precisely, given   one is interested in curves   such that:

 

for which local solutions are guaranteed by the Existence and Uniqueness Theorem of Ordinary Differential Equations. If   is furthermore a complete vector field, then the flow of  , defined as the collection of all integral curves for  , is a diffeomorphism of  . The flow   given by   is in fact an action of the additive Lie group   on  .

Conversely, every smooth action   defines a complete vector field   via the equation:

 

It is then a simple result[2] that there is a bijective correspondence between   actions on   and complete vector fields on  .

In the language of flow theory, the vector field   is called the infinitesimal generator.[3] 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  .

Definition

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Let   be a Lie group with corresponding Lie algebra  . Furthermore, let   be a smooth manifold endowed with a smooth action  . Denote the map   such that  , called the orbit map of   corresponding to  .[4] For  , the fundamental vector field   corresponding to   is any of the following equivalent definitions:[2][4][5]

  •  
  •  
  •  

where   is the differential of a smooth map and   is the zero vector in the vector space  .

The map   can then be shown to be a Lie algebra homomorphism.[5]

Applications

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Lie groups

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The Lie algebra of a Lie group   may be identified with either the left- or right-invariant vector fields on  . It is a well-known result[3] that such vector fields are isomorphic to  , the tangent space at identity. In fact, if we let   act on itself via right-multiplication, the corresponding fundamental vector fields are precisely the left-invariant vector fields.

Hamiltonian group actions

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In the motivation, it was shown that there is a bijective correspondence between smooth   actions and complete vector fields. Similarly, there is a bijective correspondence between symplectic actions (the induced diffeomorphisms are all symplectomorphisms) and complete symplectic vector fields.

A closely related idea is that of Hamiltonian vector fields. Given a symplectic manifold  , we say that   is a Hamiltonian vector field if there exists a smooth function   satisfying:

 

where the map   is the interior product. This motivatives the definition of a Hamiltonian group action as follows: If   is a Lie group with Lie algebra   and   is a group action of   on a smooth manifold  , then we say that   is a Hamiltonian group action if there exists a moment map   such that for each:  ,

 

where   and   is the fundamental vector field of  

References

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  1. ^ Hou, Bo-Yu (1997), Differential Geometry for Physicists, World Scientific Publishing Company, ISBN 978-9810231057
  2. ^ a b Ana Cannas da Silva (2008). Lectures on Symplectic Geometry. Springer. ISBN 978-3540421955.
  3. ^ a b Lee, John (2003). Introduction to Smooth Manifolds. Springer. ISBN 0-387-95448-1.
  4. ^ a b Audin, Michèle (2004). Torus Actions on Symplectic manifolds. Birkhäuser. ISBN 3-7643-2176-8.
  5. ^ a b Libermann, Paulette; Marle, Charles-Michel (1987). Symplectic Geometry and Analytical Mechanics. Springer. ISBN 978-9027724380.