---- > [!definition] Definition. ([[derived and central series of a Lie algebra]]) > Let $\mathfrak{g}$ be a [[Lie algebra]]. Let $[I,J]=\text{span}\{ [i,j] \}_{i \in I}^{j \in J}$; this is an [[ideal of a Lie algebra|ideal]].[^1] The **derived subalgebra** [[Lie subalgebra|of]] $\mathfrak{g}$ is defined to be $[\mathfrak{g}, \mathfrak{g}]$. Note that $[\mathfrak{g}, \mathfrak{g}]=0$ iff $\mathfrak{g}$ is [[abelian Lie algebra|abelian]]. > Now, fix notation $\mathfrak{g}^{(0)}=\mathfrak{g}^{0}=\mathfrak{g}$, and then inductively define ideals $\begin{align} \mathfrak{g}^{(n)}&:=[\mathfrak{g}^{(n-1)}, \mathfrak{g}^{(n-1)}] \\ \mathfrak{g}^{n} &:= [\mathfrak{g}, \mathfrak{g}^{n-1}]. \end{align}$ The **derived series** of $\mathfrak{g}$ is $\mathfrak{g} \supset \overbrace{ \mathfrak{g}^{(1)} }^{ [\mathfrak{g}, \mathfrak{g}] } \supset \overbrace{ \mathfrak{g}^{(2)} }^{ \big[ [\mathfrak{g}, \mathfrak{g}], [\mathfrak{g}, \mathfrak{g}] \big] } \supset \mathfrak{g}^{(3)} \supset\cdots $ The **central series** of $\mathfrak{g}$ is $\mathfrak{g} \supset \underbrace{ \mathfrak{g}^{1} }_{ [\mathfrak{g}, \mathfrak{g}] } \supset \underbrace{ \mathfrak{g}^{2} }_{ \big[ \mathfrak{g}, [\mathfrak{g, \mathfrak{g}}] \big] } \supset \mathfrak{g}^{3} \supset \cdots$ Note that number of derived series brackets grows exponentially, while the number of central series brackets grows linearly. ^definition-1 > [!note] Mnemonic. > Everything in the derived series is derived subalgebra. ^note > [!definition] Definition. (Nilpotent, Solvable) > We call $\mathfrak{g}$ **solvable** if its derived series terminates: $\mathfrak{g}^{(n)}=0$ for some $n \geq 1$. > We call $\mathfrak{g}$ **nilpotent** if its central series terminates: $\mathfrak{g}^{n}=0$ for some $n \geq 1$. > $\mathfrak{g}^{(n)} \subset \mathfrak{g}^{n}$, so nilpotent $\implies$ solvable.[^2] ^definition-2 > [!basicexample] > - [ ] check as a direct computation using elementary matrices: $[\mathfrak{gl}_{n}, \mathfrak{gl}_{n}]=\mathfrak{sl}_{n}$ > - Any [[abelian Lie algebra]] is nilpotent (hence solvable), because $\mathfrak{g}^{1}=[\mathfrak{g}, \mathfrak{g}]=0$. > - The [[strictly triangular matrix|Lie algebra of strictly upper triangular matrices]] $\mathfrak{n}_{n} \subset \mathfrak{gl}_{n}$ is nilpotent (hence solvable), as $\mathfrak{g}^{n}=0$ > - The [[upper-triangular matrix|upper triangular Lie algebra]] $\mathfrak{b}_{n} \subset \mathfrak{gl}_{n}$ is solvable, but not nilpotent if $n \geq 2$. ^basic-example - [ ] see also handwritten notes from lecture 6 ---- #### [^1]: Check using Jacobi identity. [^2]: The converse is false! For a counterexample, let $\mathfrak{g}$ be the two-dimensional [[Lie algebra]] with [[basis]] $x,y$ and bracket determined by $[x,y]=y$. Then $\mathfrak{g}$ is solvable but not nilpotent. ---- #### 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 > ```