# Proving that the adjoint representation of a simple Lie Algebra satisfying $Tr(T_aT_b)=\lambda \delta_{ab}$ is irreducible

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.

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 $$\mathrm{i}f_{abc}$$.