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How to find the maximal solvable Lie subalgebra of $\mathfrak sl(n,\mathbb R)$?

Maybe the invariance lemma is the key!

Ronald
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Note that a maximal solvable subalgebra need not be unique. For simplicity take $n=2$. So in $\mathfrak{sl}(2,\mathbb{R})$ we have several maximal solvable subalgebras, e.g., $\mathfrak{a}=\langle h,x\rangle$, or $\mathfrak{b}=\langle h,y\rangle$. Here $(x,y,z)$ is a basis with brackets $[x,y]=h,\,[[h,x]=2x,\,[h,y]=2y$. In general, a maximal solvable subalgebra is called a Borel subalgebra. Over the complex numbers they a re all conjugated. The so-called standard Borel subalgebra can be constructed via the weight space decomposition:

Proposition Let $L$ be a semisimple Lie algebra, and $H$ be a Cartan subalgebra with root system $\Phi$ and base $\Delta$. Then the standard Borel subalgebra is given by $$ B:=B(\Delta)=H\oplus \bigoplus_{\alpha\in \Phi^+} L_{\alpha}. $$ In terms of matrices, the standard Borel subalgebra of $\mathfrak{sl}(n)$ consists of the solvable Lie subalgebra of all upper-triangular matrices in $\mathfrak{sl}(n)$.

Dietrich Burde
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  • Thanks Dietrich, actually I am still not familiar with the root system. I am sorry I asked a very general question about maximal solvable subalgebras which I learned from you that they are not unique.. Can I ask a specific question that: prove that $t(n, F) ∩ sl(n, F)$ is a maximal solvable subalgebra of $sl(n, F)$? – Ronald Feb 19 '16 at 16:58
  • I can add this specific question to the main question and you may add the answer to your answer and we get a complete answer – Ronald Feb 19 '16 at 17:34
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    $t(n,F)\cap \mathfrak{sl}(n,F)$ just consists of all traceless upper-triangular matrices, hence is the standard Borel subalgebra. So it is maximal solvable (sorry for using again Lie algebra theory, but that's the way you should try it, I think). – Dietrich Burde Feb 19 '16 at 19:31
  • I think I will. Do you have a good and simple reference about this.. Like the link you provided about Heisenberg Algebras? – Ronald Feb 19 '16 at 19:34
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    Hmm, yes - my lecture notes. OK, this was a joke. You want this in Hindi, I guess, not in German. But of course, there are nice lecture notes in English, too. – Dietrich Burde Feb 19 '16 at 19:37
  • No it's fine,, I know some German :-) – Ronald Feb 19 '16 at 19:39
  • By the way, there is a hint on this question: Use $H$ to produce, out of opposite weights, a copy of $sl(2, F)$ inside a supposed larger subalgebra. .. But I didn't understand it! – Ronald Feb 19 '16 at 19:44