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I'm reading Georgi's text about Lie algebra, 2nd edition.

In chap 6, he introduced "Roots and Weights". What I didn't understand is the discussion of section 6.2 about the adjoint representation. He said: "The adjoint representation, is particularly important. Because the rows and columns of the matrices defined by [tex][T_a]_{bc} = -if_{abc}[/tex] are labeled by the same index that labels the generators, the states of the adjoint representation correspond to the generators themselves."

The sentence with underline is the point that I didn't understand. Why states of the adjoint representation correspond to the generators? And then he denotes the state correspond to an arbitrary generator [tex]X_a[/tex] as [tex]|X_a\rangle[/tex], moreover,

[tex]\alpha|X_a\rangle + \beta|X_b\rangle = |\alpha X_a + \beta X_b\rangle[/tex]

Could anybody show me why any state in the adjoint representation would correspond to a generator? Thanks a lot!

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# Adjoint representation correspondence?

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