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Reading some notes on Lie algebras now.

Let $g$ be the Lie algebra $sl_2(\mathbb{C}))$ and let $U$ be the universal enveloping algebra.

In the course of some computation (proving the Casimir element $C = EF + FE + H^2 / 2$ to be central), the author writes:

$[E,EF] = [E,E]F + E[E,F]$.

It's not clear to me why this is true. Is it a general fact in the universal enveloping algebra of a lie algebra that $[E,FG] = [E,F]G + E[F,G]$? Or maybe something else? Or is the author using something about $sl_2(\mathbb{C})$?

I can compute: $[E,FG] = EFG - FGE$ and $[E,F]G + E[F,G] = EFG - FEG + EFG - EGF$, which doesn't do what we want.

I can also write $[E,FG] = [E, GF + [F,G]] = [E,GF] + [E,[F,G]]$, which doesn't get us anywhere.

Elle Najt
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1 Answers1

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As your computations show, it's not true in general. Nevertheless, in this particular case: $[E, E]F + E [E, F] = EEF - EEF + EEF - EFE = EEF - EFE = [E, EF]$.

Alex M.
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  • Also, note that the formally correct (but otherwise not true) generalization would have been $[E, FG] = [E, F]G + F [E, G]$ (i.e. you would expect ${\rm ad}_E$ to act like a derivative, even though this turns out to be false). – Alex M. Jan 21 '16 at 15:51