Linear Lie algebra
In algebra, a linear Lie algebra is a subalgebra of the Lie algebra 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
Let V be a finite-dimensional vector space over a field of characteristic zero and a subalgebra of. Then V is semisimple as a module over if and only if it is a direct sum of the center and a semisimple ideal and the elements of the center are diagonalizable.
