Given a basis ei of the Lie algebra g, the matrix elements of the Killing form are given by
B^{ij}= tr (\textrm{ad}(e^i)\circ \textrm{ad}(e^j)) / I_{ad}
where Iad is the Dynkin index of the adjoint representation of g.
Here
\left(\textrm{ad}(e^i) \circ \textrm{ad}(e^j)\right)(e^k)= [e^i, [e^j, e^k]] = {c^{im}}_{n} {c^{jk}}_{m} e^n
in Einstein summation notation and so we can write
B^{ij} = \frac{1}{I_{\textrm{ad}}} {c^{im}}_{n} {c^{jn}}_{m}
where the {c^{ij}}_{k} are the structure constants of the Lie algebra. The Killing form is the simplest 2-tensor that can be formed from the structure constants.
In the above indexed definition, we are careful to distinguish upper and lower indexes (co- and contra-variant indexes). This is because, in many cases, the Killing form can be used as a metric tensor on a manifold, in which case the distinction becomes an important one for the transformation properties of tensors. When the Lie algebra is semisimple, its Killing form is nondegenerate, and hence can be used as a metric tensor to raise and lower indexes. In this case, it is always possible to choose a basis for g such that the structure constants with all upper indexes are completely antisymmetric.
B^{ij}= tr (\textrm{ad}(e^i)\circ \textrm{ad}(e^j)) / I_{ad}
where Iad is the Dynkin index of the adjoint representation of g.
Here
\left(\textrm{ad}(e^i) \circ \textrm{ad}(e^j)\right)(e^k)= [e^i, [e^j, e^k]] = {c^{im}}_{n} {c^{jk}}_{m} e^n
in Einstein summation notation and so we can write
B^{ij} = \frac{1}{I_{\textrm{ad}}} {c^{im}}_{n} {c^{jn}}_{m}
where the {c^{ij}}_{k} are the structure constants of the Lie algebra. The Killing form is the simplest 2-tensor that can be formed from the structure constants.
In the above indexed definition, we are careful to distinguish upper and lower indexes (co- and contra-variant indexes). This is because, in many cases, the Killing form can be used as a metric tensor on a manifold, in which case the distinction becomes an important one for the transformation properties of tensors. When the Lie algebra is semisimple, its Killing form is nondegenerate, and hence can be used as a metric tensor to raise and lower indexes. In this case, it is always possible to choose a basis for g such that the structure constants with all upper indexes are completely antisymmetric.
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