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Could you give me a hint how to prove the following? A representation $R$ of a complex Lie algebra $\mathfrak{g}$ is irreducible iff the image $R(\mathfrak{g})$ is a simple Lie algebra.

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What tools do you have at your disposal? Is it obvious to you that $R(\mathfrak{g})$ is a reductive Lie algebra? – Plop May 10 '11 at 9:37
Basically, I wanted to start from the definitions (apart from Schur's Lemma). To me it's not obious that $R(\mathfrak{g})$ is reductive. I thought a bit about proving the converse direction. But is that really true? My idea for a counterexample was to take a simple matrix Lie algebra $\mathfrak{g}$ and use as a reducible representation a map that embeds the matrices of $\mathfrak{g}$ into bigger matrices. Then as far as I could see also $R(\mathfrak{g})$ is simple (not abelian, no proper ideals). Where am I mistaken? – Jim May 10 '11 at 13:36
Yes, the "if" part is not true: if you take any $\mathfrak{g}$, the direct sum of two copies of the adjoint representation is never irreducible, and in that case $R(\mathfrak{g}) \simeq \mathfrak{g}/ Z(\mathfrak{g})$, so if you take a simple Lie algebra $\mathfrak{g}$ you get a counterexample. – Plop May 10 '11 at 14:23
"Schur's Lemma" tells me that you assume the Lie algebra and the representations are complex, I edited your question accordingly. Is that correct? – Plop May 10 '11 at 14:24
The exercise is completely false. Considering that the summary above the exercise is also false (it states that the adjoint representation is faithful, with no hypothesis), I suggest you close this book, and find another one. – Plop May 10 '11 at 15:23
up vote 5 down vote accepted

Here are more details about the aforementioned counterexample. Let $\mathfrak{g} = \mathfrak{sl}_2 \times \mathfrak{sl}_2$ (over $\mathbb{C}$), and consider the representation $R : \mathfrak{g} \rightarrow \mathfrak{gl}_4$, $\left( \begin{pmatrix} a & b \\ c & -a \end{pmatrix}, \begin{pmatrix} d & e \\ f & -d \end{pmatrix} \right) \mapsto \begin{pmatrix} a+d & b & e & 0 \\ c & -a+d & 0 & e \\ f & 0 & a-d & b \\ 0 & f & c & -a-d \end{pmatrix}$.

You can check directly that it is a representation, but this follows from a general construction: if $R_1$ is a representation of $\mathfrak{g}_1$ and $R_2$ is a representation of $\mathfrak{g}_2$, there is a representation $R = R_1 \otimes R_2$ of $\mathfrak{g}_1 \times \mathfrak{g}_2$ (acting on the tensor product of the underlying vector spaces $V_1$ and $V_2$) given by $R(x_1,x_2) = R_1(x_1) \otimes \mathrm{Id}_{V_2} + \mathrm{Id}_{V_1} \otimes R_2(x_2)$. Beware that there is also the notion of tensor product of two representations of the same Lie algebra.

The representation $R$ is clearly faithful, and it is not hard to show that it is irreducible (either directly in this case, are more generally show that $R_1 \otimes R_2$ is irreducible iff $R_1$ and $R_2$ are irreducible, by considering $R$ as a representation of $\mathfrak{g}_1$ and $\mathfrak{g}_2$ separately).

As to references, it depends on your profile. Are you more interested in physics or just the math? Do you want to study representations of Lie groups, or just Lie algebras (which is a prerequisite to the former)? Do you want to be thorough, or just understand the key facts of the theory in order to be able to apply it in particular cases? In any case, here are some references I know:

  • Fulton, Harris, Representation Theory: A First Course (many examples)
  • Bourbaki, Lie Groups and Lie algebras, chapter I (thorough, math, most of the basic things there is to know about nilpotent, solvable and semi-simple Lie algebras, but does not cover the classification of semi-simple Lie algebras or their representations: this is in chapters VI-IX. Read this only if you have time and love the subject)
  • Serre, Complex Semisimple Lie Algebras (concise and clear, mainly about the classification of semi-simple Lie algebras and their representations)

The last one is the best IMO.

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