From killing equation to Lie algebra

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SUMMARY

The discussion focuses on deriving the generators of Lie algebra from the Killing equation. Specifically, it addresses how to obtain the commutation relations (16) using the Killing vector (14) and its constraint (15). The participants emphasize that the full set of Killing vector fields, when combined with constant coefficients, forms a Lie algebra under the Lie bracket operation defined on the manifold. The key takeaway is that commutation relations can be derived by taking Lie brackets of the different Killing vectors.

PREREQUISITES
  • Killing equation in differential geometry
  • Lie algebra and its properties
  • Lie bracket operation
  • Understanding of vector fields on manifolds
NEXT STEPS
  • Study the derivation of commutation relations in Lie algebras
  • Learn about the properties of Killing vector fields
  • Explore the application of Lie brackets in differential geometry
  • Investigate the relationship between Killing vectors and symmetries of manifolds
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This discussion is beneficial for mathematicians, physicists, and students studying differential geometry, particularly those interested in the applications of Lie algebras and Killing equations in theoretical physics.

Esmaeil
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Hi,
I want to know how can we arrive at the generators of Lie algebra if we have killing equation ?
on the other words, In this attached image I want to know how can I arrive at communication relation (16) by starting from killing vector (14) and its constraint (15)?https://www.physicsforums.com/images/attach/jpg.gif
 

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I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?
 
If you have the full set of killing vector fields, these vector fields (with constant coefficients) form a Lie algebra under the Lie bracket operation defined on the manifold...

You should be able to get the commutation relations just by taking lie brackets of the different killing vectors. Is this operation not working for you?
 

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