- #1

CAF123

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## Main Question or Discussion Point

Given that ##g T_a g^{-1} = D^b_a T_b## one can show that the generators in the adjoint representation of a group ##G## are the structure constants of the lie algebra satisfied by the ##T_a##.

Write ##g## infinitesimal, so that ##g = 1 + \mathrm {i} \alpha^a T_a## and ##D^c_a = \delta^c_a + i \alpha_b (A_b)^c_a\,\,\,(*)## Then $$gT_ag^{-1} = T_a + i \alpha^b (T_a T_b-T_bT_a) = T_c (\delta^c_a + i \alpha^b i c^c_{ab})$$ Comparing with ##(*)## we see that ##(A_b)^c_a = i c^c_{ba}## as required.

While the manipulations are clear, I am a bit confused as to what all the components mean -

1) Is ##g## here a representation of a group element ##g \in G## or a group element itself? Since it is expanded in terms of the lie algebra I think it could be either with the generators ##T_a## in the appropriate representation?

2) ##D^c_a## is the adjoint representation of a group element so for ##SU(N)## for example would be a ##N^2-1 \times N^2-1## matrix. On the rhs of my equation, ##g T_a g^{-1} = D^b_a T_b##, this is acting on ##T_b## so would this not mean that ##T_b## is also in the adjoint representation to make the matrix multiplication make sense and therefore that ##g## itself is also an adjoint representation of ##G##?

What is the fault in this reasoning?

Thanks!

Write ##g## infinitesimal, so that ##g = 1 + \mathrm {i} \alpha^a T_a## and ##D^c_a = \delta^c_a + i \alpha_b (A_b)^c_a\,\,\,(*)## Then $$gT_ag^{-1} = T_a + i \alpha^b (T_a T_b-T_bT_a) = T_c (\delta^c_a + i \alpha^b i c^c_{ab})$$ Comparing with ##(*)## we see that ##(A_b)^c_a = i c^c_{ba}## as required.

While the manipulations are clear, I am a bit confused as to what all the components mean -

1) Is ##g## here a representation of a group element ##g \in G## or a group element itself? Since it is expanded in terms of the lie algebra I think it could be either with the generators ##T_a## in the appropriate representation?

2) ##D^c_a## is the adjoint representation of a group element so for ##SU(N)## for example would be a ##N^2-1 \times N^2-1## matrix. On the rhs of my equation, ##g T_a g^{-1} = D^b_a T_b##, this is acting on ##T_b## so would this not mean that ##T_b## is also in the adjoint representation to make the matrix multiplication make sense and therefore that ##g## itself is also an adjoint representation of ##G##?

What is the fault in this reasoning?

Thanks!