Ado's theorem

Ado's theorem states that every finite-dimensional Lie algebra over a field has a faithful finite-dimensional representation by linear transformations.

Lie algebra: A vector space with a bilinear, alternating bracket satisfying the Jacobi identity. Ado's theorem applies to every finite-dimensional Lie algebra.
Adjoint representation: The representation of a Lie algebra on itself by brackets with its elements. It gives a natural first action, though its kernel is the center.
Igor Shafarevich: A Soviet mathematician known for major work in algebraic number theory and algebraic geometry. His name is sometimes associated with a separate theorem, not Ado's result.
Matrix Lie algebra: A vector space of matrices closed under the commutator bracket. Ado's theorem realizes every finite-dimensional Lie algebra in this concrete form.
Faithful representation: A representation whose kernel is zero, so distinct algebra elements act distinctly. Faithfulness is the key guarantee in Ado's theorem.
Center of a Lie algebra: The subalgebra of elements that bracket to zero with every element of a Lie algebra. The adjoint representation misses precisely these central elements.
Ado–Iwasawa theorem: A name for Ado's theorem in formulations emphasizing its extension or proof by Kenjiro Iwasawa. The attribution reflects Iwasawa's contribution to the theorem's development.
Lie group: A group that is also a smooth manifold, with smooth multiplication and inversion. Finite-dimensional Lie algebra representations connect to linear actions of Lie groups.
Universal enveloping algebra: An associative algebra that encodes a Lie algebra while preserving its bracket as a commutator. Its representations provide a route from Lie algebra actions to linear transformations.
Nilpotent Lie algebra: A Lie algebra whose lower central series eventually becomes zero. Ado's theorem is often proved first for nilpotent Lie algebras.

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