Basis for lie algebra sl(n,F) : $\mathfrak{sl}(2,F)$

lie-algebraslie-groupsproblem solvingsemisimple-lie-algebras

$\mathfrak{sl}(2,F)$ is lie subalgebra of $\mathfrak{gl}(2,F)$.
Let $x=\begin{pmatrix} 0 &1\\ 0&0 \end{pmatrix}$, $h=\begin{pmatrix} 1 &0\\ 0&-1 \end{pmatrix}$
$y=\begin{pmatrix} 0 &0\\ 1&0 \end{pmatrix}$.

I understand $\{x,y,z \}$ forms a basis for $\mathfrak{sl}(2,F)$ and had found the structure constants.

I have following urging questions. Please help me with this.

  1. Why people specifically working on this basis?
  2. Is there a similar ordered basis for $\mathfrak{sl}(n,F)$?
  3. Suggest me some good set of problems on basic lie algebra to practice.

Thank you in advance.

Best Answer

Question 1: This basis fits to the general theory of semisimple Lie algebras. Namely, $x$ and $y$ are nilpotent, with $x\in \mathfrak{n}^+$ and $y\in \mathfrak{n}^-$, for $\mathfrak{n}$ the nilradical of the standard Borel subalgebra, and $h\in \mathfrak{h}$ is diagonal. So $$\mathfrak{sl}_2(F)=\mathfrak{n}^-\oplus \mathfrak{h}\oplus \mathfrak{n}^+$$

Question 2: Yes, there is a similar one for $\mathfrak{sl}_n(F)$, where the diagonal matrices span the Cartan subalgebra $\mathfrak{h}$.

Question 3: Indeed, take the problems of Humphrey's book. Solutions are even online (see for example here).

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