linear Lie algebra

In algebra, a linear Lie algebra is a subalgebra $\backslash mathfrak\{g\}$ of the Lie algebra $\backslash mathfrak\{gl\}(V)$ consisting of endomorphisms of a vector space V. In other words, a linear Lie algebra is the image of a Lie algebra representation.

Any Lie algebra is a linear Lie algebra in the sense that there is always a faithful representation of $\backslash mathfrak\{g\}$ (in fact, on a finite-dimensional vector space by Ado's theorem if $\backslash mathfrak\{g\}$ is itself finite-dimensional.)

Let V be a finite-dimensional vector space over a field of characteristic zero and $\backslash mathfrak\{g\}$ a subalgebra of $\backslash mathfrak\{gl\}(V)$. Then V is semisimple as a module over $\backslash mathfrak\{g\}$ if and only if (i) it is a direct sum of the center and a semisimple ideal and (ii) the elements of the center are diagonalizable (over some extension field).{{harvnb|Jacobson|1979|loc=Ch III, Theorem 10}}