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I am new to StackExchange. I am learning about Lie-Algebras and I was wondering whether somebody can give me an example for a finite dimensional and faithful representation of a sovable and finite dimensional Lie-Algebra $\mathcal{L}$ over a Vektorspace V (finite dimensional): $\rho: \mathcal{L} \longrightarrow gl(V)$, such that if $\mathcal{N}$ is the Nil-Radical of $\mathcal{L}$, $~ \rho(\mathcal{N})$ is nilpotent.

Such a representation should exists (see: Theorem 9 of https://terrytao.wordpress.com/2011/05/10/ados-theorem/). I can think of one for a nilpotent Lie-Algebra, but I have trouble finding one for a sovable Lie-Algebra.

Thank you! :)

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  • $\begingroup$ You are right, thanks. I misused the term "Nil-representation". I changed the question. $\endgroup$ – Stephi Feb 26 at 13:13
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Let $\mathfrak{r}_2(K)$ be the non-abelian Lie algebra of dimension $2$ over $K$ with basis $(e_1,e_2)$ and $[e_1,e_2]=e_1$. Then the adjoint representation is a faithful linear representation of $\mathfrak{r}_2(K)$ into $\mathfrak{gl}_2(K)$ and the nilradical $\mathfrak{n}=\langle e_1\rangle $ is mapped to the nilpotent matrix $$ {\rm ad}(e_1)=\begin{pmatrix} 0 & 1 \cr 0 & 0 \end{pmatrix}. $$ The adjoint representation is faithful in this case, because its kernel, the center of the Lie algebra, is trivial. This is never the case for non-abelian nilpotent Lie algebras.

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What about the natural action of the Lie algebra of upper triangular $n\times n$ matrices on $k^n$? The nilradical consists of strictly upper triangular matrices, which are nilpotent.

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  • $\begingroup$ Then we need first to represent our given solvable Lie algebra as a subalgebra of upper-triangualr matrices. So do you want to apply Lie's Theorem? This applies for algebraically closed fields of characteristic zero. $\endgroup$ – Dietrich Burde Feb 26 at 13:55
  • $\begingroup$ @Dietrich Burde He just asked for an example, I don't claim that this is a proof for all solvable Lie algebras $\endgroup$ – Antoine Labelle Feb 26 at 14:00
  • $\begingroup$ I thought "example for a finite-dimensional faithful representation of a (given) solvable Lie algebra". Then a minimal example in dimension $2$ already works. $\endgroup$ – Dietrich Burde Feb 27 at 9:19

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