Lie's theorem

In mathematics, specifically the theory of Lie algebras, Lie's theorem states that,^[1] over an algebraically closed field of characteristic zero, if ${\displaystyle \pi :{\mathfrak {g}}\to {\mathfrak {gl}}(V)}$ is a finite-dimensional representation of a solvable Lie algebra, then there's a flag ${\displaystyle V=V_{0}\supset V_{1}\supset \cdots \supset V_{n}=0}$ of invariant subspaces of ${\displaystyle \pi ({\mathfrak {g}})}$ with ${\displaystyle \operatorname {codim} V_{i}=i}$, meaning that ${\displaystyle \pi (X)(V_{i})\subseteq V_{i}}$ for each ${\displaystyle X\in {\mathfrak {g}}}$ and i.

Put in another way, the theorem says there is a basis for V such that all linear transformations in ${\displaystyle \pi ({\mathfrak {g}})}$ are represented by upper triangular matrices.^[2] This is a generalization of the result of Frobenius that commuting matrices are simultaneously upper triangularizable, as commuting matrices generate an abelian Lie algebra, which is a fortiori solvable.

A consequence of Lie's theorem is that any finite dimensional solvable Lie algebra over a field of characteristic 0 has a nilpotent derived algebra (see #Consequences). Also, to each flag in a finite-dimensional vector space V, there correspond a Borel subalgebra (that consist of linear transformations stabilizing the flag); thus, the theorem says that ${\displaystyle \pi ({\mathfrak {g}})}$ is contained in some Borel subalgebra of ${\displaystyle {\mathfrak {gl}}(V)}$.^[1]