Linear Lie algebra

In algebra, a linear Lie algebra is a subalgebra ${\displaystyle {\mathfrak {g}}}$ of the Lie algebra ${\displaystyle {\mathfrak {gl}}(V)}$ 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 ${\displaystyle {\mathfrak {g}}}$ (in fact, on a finite-dimensional vector space by Ado's theorem if ${\displaystyle {\mathfrak {g}}}$ is itself finite-dimensional.)

Let V be a finite-dimensional vector space over a field of characteristic zero and ${\displaystyle {\mathfrak {g}}}$ a subalgebra of ${\displaystyle {\mathfrak {gl}}(V)}$. Then V is semisimple as a module over ${\displaystyle {\mathfrak {g}}}$ 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).¹