Hi guys! I am getting some sort of contradiction using the definition of the killing-form.(adsbygoogle = window.adsbygoogle || []).push({});

The killing form as a matrix (sometimes called metric) in some basis can be written as:

[tex] \eta_{ab}=f_{ac}^df_{bd}^c [/tex]

where [ itex ]

f_{ab}^c [ /itex ]

are the structure constants of the Lie algebra. Of course, a,b,c=1...dim(G), where dim(G) is the total number of the elements of the basis of the Lie algebra (the number of independent generators).

So, [ itex ]

\eta [ /itex ] is a [ itex ] dim(G)\times dim(G) [ /itex ]

matrix.

What it confuses me is that, for example, in SO(3,1) the metric tensor is [ itex ]

\eta_{ab}=diag(-1,+1,+1,+1) [ /itex ]

. It has dimension: [itex] 4\times 4 [/itex] . But if I use the definition above of the Killing-form, I get a matrix with dimension 6\times 6 , because the SO(3,1) has 6 generators..

What I am doing wrong ? or there's some concept wrong, i dont know.

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# Question on the dimension of Killing-form

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