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I am studying Humphreys' book Introduction to Lie Algebras and Representation Theory and in it, I came across this: "In fact, ad $x \in$ Der $L$, because we can rewrite the Jacobi identity in the form: $[x[yz]] = [[xy]z]+[y[xz]].$ Derivations of this form are called inner, all other outer." Is ad $x$ the only kind of inner derivation? Can you give a concrete example of an outer derivation and why is it not an inner derivation?

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  • $\begingroup$ It's hard to believe that one can find such a incoherent definition in a published book, when one then defines the Lie algebra of outer derivations to be the quotient of the Lie algebra of derivations by its ideal of inner derivations. Coherent terminology is to simply call derivations that are not inner "non-inner derivations". $\endgroup$
    – YCor
    May 28 '17 at 9:15
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Consider, for example, all derivations of the Heisenberg Lie Algebra $\mathfrak{h}_3(K)$, i.e., all linear maps $D\colon \mathfrak{h}_3(K) \rightarrow \mathfrak{h}_3(K)$ satisfying $D([x,y])=[D(x),y]+[x,D(y)]$ for all $x,y$. Here the brackets are given by $[e_1,e_2]=-[e_2,e_1]=e_3$, where $(e_1,e_2,e_3)$ denotes a basis. The inner derivations are of the form $ad (x)$, and are linear combinations of $$ ad (e_1)=\begin{pmatrix} 0 & 0 & 0\cr 0 & 0 & 0\cr 0 & 1 & 0\cr \end{pmatrix},\; ad (e_2)=\begin{pmatrix} 0 & 0 & 0\cr 0 & 0 & 0\cr -1 & 0 & 0\cr \end{pmatrix},\, ad(e_3)=0. $$ However, the Heisenberg Lie Algebra has many non-inner derivations. In fact, all linear maps of the form $$ D=\begin{pmatrix} d_1 & d_4 & 0\cr d_2 & d_5 & 0\cr d_3 & d_6 & d_1+d_5\cr \end{pmatrix} $$ are derivations of the Heisenberg Lie algebra. It is easy to see that, for example, the diagonal derivation $d=(1,1,2)$ is not inner.

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Any linear map on an Abelian Lie algebra.

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  • $\begingroup$ If $\delta$ is the linear map, then shouldn't $\delta(xy)$ be 0 for it to be a derivation on an abelian lie algebra? $\endgroup$
    – Iguana
    May 27 '17 at 19:25
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    $\begingroup$ @Iguana But $[xy]=0$ in an Abelian Lie algebra. $\endgroup$ May 27 '17 at 19:28
  • $\begingroup$ Oh yes, that is true. Thanks! Also, is it true that ad $x$ is the only kind of inner derivation? $\endgroup$
    – Iguana
    May 27 '17 at 19:29
  • $\begingroup$ @Iguana But that's the definition of inner derivation: they are the $\text{ad}\,x$. $\endgroup$ May 27 '17 at 19:30
  • $\begingroup$ Okay. I was just confused because of the way the sentence was phrased in the book. $\endgroup$
    – Iguana
    May 27 '17 at 19:31

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