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I want to calculate the maximal torus of derivations for a Lie Algebra but I don't know how.

By definition:

A maximal abelian subalgebra of the derivations algebra $Derg$ constituted of the semisimple derivations is called maximal torus of derivations of $g$ (where $g$ is the Lie algebra).

The result:

If

Let $L_n$ be the $n+1-$ dimensional Lie algebra defined by: $[X_0,X_i]=X_{i+1}, i=1,...,n-1$ where $(X_0,...,X_n)$ is a basis of $L_n$.

Then

the endomorphisms $d_1$ and $d_2$ spanned the maximal torus of derivations of $L_n$, where: $d_1(X_0)=0, d_1(X_i),1\leq i\leq n$ and $d_2(X_0)=X_0, d(X_i)=(i-1)X_i,1\leq i\leq n$

My attempt:

Let $d\in Der g$ then $d=diag(a_{00},...,a_{nn})$ because it is semisimple.

In the other hand $d$ verify $d[x,y]=[dx,y]+[x,dy]$.By calculating I found $a_{ii}=a_{00}+a_{i-1,i-1}$ for $1 \leq i\leq n+1$.

so by replacing I have $d=diag(a_{00},a_{00}+a_{11},a_{00}+2a_{11},...,(n-2)a_{00}+a_{nn})$.

Here I'm stuck I don't know how to complete, and if what I did is correct.

Can someone help? And in general how to calculate the maximal torus?

Mary Maths
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  • for the definition and the result are from here: https://link.springer.com/chapter/10.1007/978-94-011-5072-9_5 – Mary Maths May 01 '23 at 09:04
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    Semisimple only means that $d$ is diagonalisable over $\Bbb C$. You cannot assume that it is already diagonal. But this follows easily from a direct calculation with your basis. Then you will also see the result. Try it first for $n=3,4,5$ and then prove it for general $n$. Of course you can find the proof also in the literature (e.g., in the book by Goze and Khakimdjanov, which is not the article you have linked). – Dietrich Burde May 01 '23 at 09:21
  • can you give how to begin I don't know @dietrichBurde – Mary Maths May 01 '23 at 13:47
  • For $n=3$ see my answer here. We have $d_1=diag(1,0,1)$ and $d_2=(0,1,1)$. – Dietrich Burde May 01 '23 at 14:58
  • @DietrichBurde: I understand your answer in the link, which consists of determining the derivation, but how we conclude for the maximal torus? the can you write the answer please? – Mary Maths May 03 '23 at 12:37
  • Determining the derivations includes determining the ones from the maximal torus, right? You can just prove the general result about all derivations of $L_n$. Since this is in the book by Goze, there is no need to copy it here. – Dietrich Burde May 03 '23 at 15:08

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