---- > [!definition] Definition. ([[derivation]]) > Let $A$ be an [[algebra]] over a [[ring]] $R$. A **derivation** on $A$, or **$A$-derivation**, is an $R$-[[linear map|linear]] morphism $d: A \to A$ such that for all $a,b \in A$ we have that the **Leibniz rule** $d(ab)=d(a)b + ad(b)$ is satisfied. > More generally, if $M$ is an $(A, A)$-[[bimodule]], an $R$-linear map $d:A \to M$ satisfying the Leibniz rule is also called a derivation. > The collection of all $R$-derivations $A \to A$ is denoted $\text{Der}_{R}(A)$. The collection of all $R$-derivations of $A$ into an $A$-[[module]] $M$ is denoted $\text{Der}_{R}(A, M)$. > [!basicexample] > >- A derivation $\delta:\mathfrak{g} \to \mathfrak{g}$ of a [[Lie algebra]] $\mathfrak{g}$ is a [[linear map]] satisfying $\delta([x,y])=[x, \delta(y)]+ [\delta(x), y]$ for all $x,y \in \mathfrak{g}$. The set $\text{Der }\mathfrak{g}$ of derivations of $\mathfrak{g}$ forms a [[Lie subalgebra]] of the [[general linear Lie algebra]] $\mathfrak{gl}(\mathfrak{g})$. > >- Let $M$ be a [[smooth manifold]]. Put $R=\mathbb{R}$, $A=C^{\infty}(M)$ in the definition. A $C^{\infty}(M)$-[[derivation]] $D:C^{\infty}(M) \to \mathbb{R}$ is an $\mathbb{R}$-[[linear map]] satisfying the Leibniz rule: for all smooth $f,g:M \to \mathbb{R}$, one has $D(fg)=D(f)g+fD(g)$. The notion of a tangent vector is subtly different: it involves the notion of a derivation *at a point*. ---- #### > [!justification] > We want to show that the set $\text{Der }\mathfrak{g}$ of derivations of $\mathfrak{g}$ forms a [[Lie subalgebra]] of the [[general linear Lie algebra]] $\mathfrak{gl}(\mathfrak{g})$. > Clearly it is a [[linear subspace]] of $\mathfrak{gl}(\mathfrak{g})$. And given $\delta_{1},\delta_{2} \in \text{Der }\mathfrak{g}$ one has for all $x,y \in \mathfrak{g}$ that >$\begin{align} ([\delta_{1}, \delta_{2}]_{\mathfrak{gl}(V)})([x,y]_{\mathfrak{g}}) &= (\delta_{1}\delta_{2} - \delta_{2} \delta_{1})([x,y]) \\ &= \delta_{1}\delta_{2}([x,y]) - \delta_{2} \delta_{1}([x,y]) \\ &= \delta_{1}\big( [x, \delta_{2}(y) ] + [\delta_{2}(x), y] \big) - \delta_{2}\big( [x, \delta_{1}(y) ] + [\delta_{1}(x), y] \big) \\ & = \textcolor{Skyblue}{\delta_{1} ( [x, \delta_{2}(y) ])} + \textcolor{Apricot}{\delta_{1}([\delta_{2}(x), y] )} \\ & \ \ - \textcolor{LimeGreen}{\delta_{2}( [x, \delta_{1}(y) ])} - \textcolor{Thistle}{\delta_{2}([\delta_{1}(x), y])} \\ &= \textcolor{Skyblue}{[x, \delta_{1} \delta_{2} (y)] + \cancel{[\delta_{1}(x), \delta_{2}(y)]} } \\ & \ \ + \textcolor{Apricot}{\cancel{[\delta_{2}(x), \delta_{1}(y)]} + [\delta_{1} \delta_{2}(x), y]} \\ & \ \ -\textcolor{LimeGreen}{ [x, \delta_{2} \delta_{1} (y)] - \cancel{[\delta_{2}(x), \delta_{1}(y)]}} \\ & \ \ - \textcolor{Thistle}{[\cancel{\delta_{1}(x), \delta_{2}(y)]} - [\delta_{2}\delta_{1}(x), y]} \\ &= ( \textcolor{Skyblue}{[x, \delta_{1} \delta_{2} (y)]} \textcolor{LimeGreen}{- [x, \delta_{2}\delta_{1}(y)]} ) \\ & \ \ + ( \textcolor{Apricot}{[\delta_{1}\delta_{2}(x), y]} \textcolor{Thistle}{- [\delta_{2} \delta_{1} (x), y]} ) \\ & = [x, (\delta_{1} \delta_{2} - \delta_{2} \delta_{1} )(y)] + [(\delta_{1}\delta_{2} - \delta_{2} \delta_{1})(x), y] \\ & = [x, ([\delta_{1}, \delta_{2}])(y)] + [ ([\delta_{1}, \delta_{2}])(x), y ] \end{align}$ thus $[\delta_{1},\delta_{2}] \in \text{Der }\mathfrak{g}$ and so $\text{Der }\mathfrak{g}$ indeed stable under the bracket, as required. ^justification ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```