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Let $k$ be a field and $\mathfrak{g}$ a Lie algebra over $k$. Put $K(\mathfrak{g}) = \bigcap_{f\in\mathrm{Der}(\mathfrak{g})} \mathrm{Ker}(f)$, which is a Lie subalgebra of $\mathfrak{g}$.

Question. Is it true that $K(\mathfrak{g})=0$ for any (finite-dimensional) $\mathfrak{g}$? If not, is there some characterization?

Here are my quick observations:

  • If $\mathfrak{g}$ is finite-dimensional and semisimple, then $K(\mathfrak{g})=0$.
  • If $\mathfrak{g}$ is positively graded (e.g. abelian), then $K(\mathfrak{g})=0$.
  • If $K(\mathfrak{g_i})=0$, then $K\left(\bigoplus_i\mathfrak{g}_i\right) = 0$.

I haven't found any counterexamples yet. Any comments are appreciated.

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    $\begingroup$ Some observations: by considering outer derivations we know $K(\mathfrak g)\subset\mathfrak z(\mathfrak g)$, the center of $\mathfrak g$. So let's assume $\mathfrak z(\mathfrak g)\ne0$. Then any linear homomorphism $\mathfrak g\to \mathfrak z(\mathfrak g)$ which is zero on the commutator $[\mathfrak g,\mathfrak g]$ is a derivation, so $K(\mathfrak g)\subset [\mathfrak g,\mathfrak g]$ as well. $\endgroup$
    – Kenta Suzuki
    Commented Nov 27 at 18:23
  • $\begingroup$ Also: Togo's 1966 paper Outer derivations of Lie algebras may be relevant. $\endgroup$
    – Kenta Suzuki
    Commented Nov 27 at 18:27
  • $\begingroup$ @KentaSuzuki, outer $\to$ inner in your first comment, right? $\endgroup$
    – LSpice
    Commented Nov 29 at 3:01
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    $\begingroup$ @LSpice yes that was a typo, my bad! $\endgroup$
    – Kenta Suzuki
    Commented Nov 29 at 3:11

1 Answer 1

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A class of counterexamples is given by the so-called characteristically nilpotent Lie algebras. For a Lie algebra $\mathfrak{g}$, consider the central descending chain defined recursively by

$$\mathfrak{g}^{[1]}=\operatorname{Der}(\mathfrak{g})(\mathfrak{g})=\{X\in \mathfrak{g}\vert \, X=f(Y), f\in \operatorname{Der}(\mathfrak{g}), Y\in \mathfrak{g} \}; $$

$$\mathfrak{g}^{[k]}=\operatorname{Der}(\mathfrak{g})(\mathfrak{g}^{[k-1]}), \; \; k>1.$$

The Lie algebra $\mathfrak{g}$ is said to be characteristically nilpotent if $\mathfrak{g}^{[m]}=0$ for some $m$ sufficiently large. In this case, it is clear that $K(\mathfrak{g})\neq 0$.

For reference, I suggest the celebrated paper

J. Dixmier, W. G. Lister: Derivations of Nilpotent Lie Algebras, Proc. Amer. Math. Soc., 8 (1957), 155–158

or the excellent survey

J. M. Ancochea, R. Campoamor: Characteristically nilpotent Lie algebras: a survey. Extracta Math. 16 (2001), 153–210.

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  • $\begingroup$ Is the 8-dimensional Lie algebra described by Diximer and Lister the minimal example? $\endgroup$
    – Kenta Suzuki
    Commented Nov 28 at 21:15
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    $\begingroup$ It is the characteristically nilpotent Lie algebra of minimal dimension, but I do not know if there exist Lie algebras of lower dimensional with $K(\mathfrak{g})\neq 0$. $\endgroup$
    – Salvatore Siciliano
    Commented Nov 28 at 23:04

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