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In some textbooks, Lie algebra is written as $$[X_a, X_b]=if_{abc}X_c$$ where $X_a$'s are the generators of the Lie group. But sometimes, the Lie algebra is written as $$[X_a,X_b]=if_{ab}^{~~~ c}X_c$$ or sometimes as $$[X^a, X^b]=if^{ab}_{~~~ c}X^c$$ where the index $c$ on the right-hand side of each expression is assumed to be summed over and $f$'s stand for structure constants.

Why are indices sometimes written as upstairs and sometimes as downstairs and sometimes everything downstairs? Is there a tool for raising or lowering indices?

Solidification
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1 Answers1

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  1. Yes, the Killing form yields a natural notion of a metric tensor for a Lie algebra. It is non-degenerate if the Lie algebra is semisimple.

  2. For compact semisimple Lie algebras many physicists use a choice of basis where the components of the metric tensor is the identity matrix, so that there are no difference between upper and lower Lie algebra indices.

Qmechanic
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