Structure constants to metric

  • Thread starter jfy4
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  • #1
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Hi,

Let's say I have a 10 dimensional Lie algebra over some field of functions, something along the lines of at least twice differentiable with twice differentiable inverses. The structure constants have inputs from this field. Is it possible to build a metric from these structure constants?

I have seen that a symmetric bi-linear form (the Killing-Cartan Form) that can also be non-degenerate for semi-simple algebras can be formed through contraction of the structure constants [itex]\kappa f^{\alpha\beta}_{\quad\gamma} f^{\delta\gamma}_{\quad\beta}=K^{\alpha\delta}[/itex]. Are there any other contractions or tensors one can form from structure constants over a field of functions (something other than [itex]\mathbb{R}[/itex] or [itex]\mathbb{C}[/itex])?

Thanks,
 

Answers and Replies

  • #2
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The field doesn't matter. What matters is, whether the Killing form is non degenerate or not, i.e. whether the Lie algebra is semisimple or not. I won't call the Killing form a metric on the structure constants, or any other attempt to define a metric on this finite set: Why? The Killing form defines a metric on the root spaces.
See: https://www.physicsforums.com/insights/lie-algebras-a-walkthrough-the-basics/
 

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