In algebra, a linear Lie algebra is a subalgebra
g
of the Lie algebra
g
l
(
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
g
(in fact, on a finite-dimensional vector space by Ado's theorem if
g
is itself finite-dimensional.)
Let V be a finite-dimensional vector space over a field of characteristic zero and
g
a subalgebra of
g
l
(
V
)
. Then V is semisimple as a module over
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).
