nLab
matrix Lie algebra

Context

\infty-Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Definition

A matrix Lie algebra is the Lie algebra that is canonically a sub-Lie algebra of the general linear algebra? Mat(n)=𝔤𝔩(n)Mat(n) = \mathfrak{gl}(n) of n×nn \times n matrices.

See also matrix Lie group.

Properties

By Ado’s theorem, every finite-dimensional Lie algebra over a field of characteristic zero is a matrix Lie algebra. A similar statement fails for Lie groups. Ado’s theorem has been used as a major step in the traditional proofs of the Cartan–Lie theorem on the existence of integration of Lie algebras to Lie groups.

Revised on September 22, 2010 04:21:06 by Toby Bartels (75.88.94.254)