- #1

tom.stoer

Science Advisor

- 5,766

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Taking an su(n) Lie algebra with hermitean generators we have

[tex][T^a, T^b] = if^{abc}T^c[/tex]

One immediately finds that the new generators

[tex]\tilde{T}^a = (-T^a)^\ast [/tex]

define the same algebra, i.e. fulfil the same commutation relations

[tex][\tilde{T}^a, \tilde{T}^b] = if^{abc}\tilde{T}^c[/tex]

One can show easily that for n=2 the two sets of generators are equivalent, i.e. related by a transformation

[tex]\tilde{T}^a = S T^a S^{-1}[/tex]

I know that for n>2 this is no longer true, i.e. that the two representations are not equivalent. That means that for n>2 this S cannot exist. My question is,

*how can one show algebraically that for n=3, 4, ... no S can exist such that*

[tex]\tilde{T}^a = S T^a S^{-1}[/tex]