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I am following the book "Lie Algebras in Particle Physics" by Howard Georgi and on page 51 he claims the statement above and goes on to prove it. I am new to this so my doubt might be trivial, but anyway here it is:

The adjoint representation of a simple Lie Algebra satisfying $Tr(T_aT_b)=\lambda\delta_{ab}$ is irreducible. To see this, assume the contrary. Then there is an invariant subspace in the adjoint representation. But the states of the adjoint representation correspond to generators, $T_r$ for $r=1$ to $K$.

In any representation we talk about operators being mapped to individual group elements. If $G$ is a Lie group then all the information regarding it is contained in the Lie Algebra of the group i.e. $$[X_a,X_b]=if_{abc}X_c$$ where $f_{abc}$ are the structure constants. We defined the adjoint representation using this as: $$[T_a,T_b]=if_{abc}T_c$$ where $$[T_a]_{bc}=-if_{abc}.$$ The questions are:

  • a. Now the author calls $T$'s as the generators instead of the $X$'s. Why is that?

  • b. What are the operators of this representation? In short how does solely $[T_a,T_b]=if_{abc}T_c$ make him see operators corresponding to some elements?

  • c. To what are those operators being mapped? Is it the group elements? Or is it somehow related to the algebra?

Any help is appreciated.

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The adjoint representation of a Lie algebra $\mathfrak{g}$ is the representation given by $$ \rho : \mathfrak{g}\mapsto\mathrm{GL}(\mathfrak{g}), k \mapsto \mathrm{ad}_k,$$ where $\mathrm{ad}_k(l) = [k,l]$ for any $k,l\in\mathfrak{g}$. In other words, it's just the representation you get when letting the Lie algebra act on itself via the Lie bracket.

If you now choose a basis of generators $X_a$ for $\mathfrak{g}$, then you can see that the matrix components of $T_a := \mathrm{ad}_{X_a}$ in this basis are exactly the structure constants $\mathrm{i}f_{abc}$, so this way of defining the adjoint representation is isomorphic to considering a representation with matrix reprsentations $[T_a]_{bc} = \mathrm{i}f_{abc}$.

As for what Georgi probably means by just defining the representation via $[T_a]_{bc} = \mathrm{i}f_{abc}$: Consider a vector space with the same dimension as the Lie algebra and pick an arbitary basis. Then $[T_a]_{bc} = \mathrm{i}f_{abc}$ defines matrices $T_a$, and you can check that their commutators are the same as the Lie bracket of the algebra, so these matrices form a representation of the Lie algebra (simply via the map $X_a\mapsto T_a$). This way of defining the adjoint representation is isomorphic to the approach via the $\mathrm{ad}_{X_a}$ above since the resulting matrices have exactly the same entries $\mathrm{i}f_{abc}$.

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