Ado’s theorem - Imported Knowlpedia Content

Let $\mathfrak{g}$ be a finite-dimensional Lie algebra over a field of characteristic $0$ (typically $\mathbb{R}$ or $\mathbb{C}$ ).

Theorem (Ado). There exists a finite-dimensional vector space $V$ and an injective Lie algebra homomorphism

\rho:\mathfrak{g}\hookrightarrow \mathfrak{gl}(V),

so $\mathfrak{g}$ is isomorphic to a Lie subalgebra of the general linear Lie algebra.

Equivalent characterizations

Equivalently, every finite-dimensional Lie algebra is a “matrix Lie algebra” up to isomorphism.

Remarks

Motivation. Ado’s theorem guarantees that Lie algebra theory can be studied inside $\mathfrak{gl}_n$ using linear algebra. It complements Lie’s third theorem, which integrates Lie algebras to (simply connected) Lie groups, by ensuring that the infinitesimal data can always be realized concretely as endomorphisms.