Equivalent characterizations of solvability for Lie algebras

Theorem (TFAE: solvability)

Let $\mathfrak g$ be a finite-dimensional Lie algebra over an algebraically closed field of characteristic $0$ (e.g. $\mathbb C$). The following are equivalent.

1. Derived series terminates: the derived series $\mathfrak g^{(0)}=\mathfrak g$, $\mathfrak g^{(k+1)}=[\mathfrak g^{(k)},\mathfrak g^{(k)}]$ satisfies $\mathfrak g^{(N)}=0$ for some $N$. Equivalently, $\mathfrak g$ is solvable .

2. Upper-triangular matrix model: there exists a faithful finite-dimensional representation (guaranteed in general by Ado’s theorem ) such that the image of $\mathfrak g$ is conjugate into the Lie algebra of upper triangular matrices in $\mathfrak{gl}(n)$; compare upper triangular examples . In particular, $\mathfrak g$ is isomorphic to a Lie subalgebra of an upper triangular matrix Lie algebra.

3. Lie’s theorem behavior in representations: for every finite-dimensional representation $\rho:\mathfrak g\to\mathfrak{gl}(V)$, there exists a complete flag of $\mathfrak g$-invariant subspaces

   $$0=V_0\subset V_1\subset \cdots \subset V_{\dim V}=V$$

   with $\dim V_i=i$. Equivalently, all operators in $\rho(\mathfrak g)$ can be simultaneously upper-triangularized. (This is the content of Lie’s theorem applied to solvable Lie algebras.)

4. Cartan’s trace criterion: $\mathfrak g$ satisfies the trace-vanishing condition equivalent to solvability given by Cartan’s criterion for solvability .

Context

Condition (1) is the intrinsic definition; (2) and (3) explain why solvable Lie algebras behave like “triangular” objects in linear algebra, while (4) is useful when $\mathfrak g$ is given abstractly but $\mathrm{ad}$ is accessible. Solvability is weaker than nilpotency, and nilpotent Lie algebras are always solvable .