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Let $L = n(n, F)$, the Lie algebra of strictly upper triangular $n \times n$ matrices over a field $F$. Show that $L_k$ has a basis consisting of all the matrix units $e_{ij}$ with $j −i > k$. Hence show that $L$ is nilpotent. What is the smallest m such that Lm = 0?

I understand that nilpotent means that when applying the bracket operator repeatedly the contents of the matrix $\rightarrow 0$ after some number of applications, but I'm a little unsure of how to approach this proof. Any help would be appreciated since I'm still very new to Lie algebras.

emonHR
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Rusk
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1 Answers1

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My hint is the following: you can try to understand what happen if you consider the bracket $ [E_{i, j}, E_{i+1,j+1}]$.

Then look at the bracket $ [E_{i, j}, E_{i+h,j+h}]$ and try to generalize the reasoning to general pairs of elements in the basis of $N(N,F)$.

I think you can then easily conclude using the fact that the elements $E_{inj}$ are a basis.

Sabino Di Trani
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