Let $\mathfrak{g}$ be a solvable Lie algebra. By Lie's theorem, it is easy to see that any finite dimensional irreducible representation is 1 dimensional. Is it possible to remove the condition that the representation be finite dimensional?
[Math] Are all irreducible representations of solvable Lie algebras 1-dimensional
lie-algebrasrepresentation-theory
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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.
Since $\mathfrak{g}$ is semisimple, it is a direct sum of simple ideals $\mathfrak{g}=\mathfrak{g}_1\oplus \cdots \oplus \mathfrak{g}_s$. By looking at the minimal dimension of a nontrivial irreducible representation of the simple Lie algebras $\mathfrak{g}_i$, we see that the dimension can be $2$ only for $\mathfrak{sl}_2(\Bbb C)$, see here, the table on page $3$. Hence one of the factors equals $\mathfrak{sl}_2(\Bbb C)$. For the irreducible representations of direct sums of simple Lie algebras see here:
What are the irreducible representations of a direct sum of Lie Algebras?
Best Answer
The link given by Qiaochu Yuan contains a counterexample if you remove finite dimensional. Consider the 3-dimensional Lie algebra (over a field $k$) generated by $x$, $d/dx$ and $1$. This is a nilpotent Lie algebra (hence solvable), with $1$ central and $$[d/dx,x]=1.$$ This Lie algebra acts on the polynomial ring $k[x]$ without eigenvalues. In fact, the enveloping algebra of this Lie algebra is the Heisenberg algebra, $H$, and $k[x]$ is a faithful irreducible $H$-module.