Linear Lie algebra explained

In algebra, a linear Lie algebra is a subalgebra

ak{g} of the Lie algebra ak{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

ak{g} (in fact, on a finite-dimensional vector space by Ado's theorem if ak{g} is itself finite-dimensional.)

Let V be a finite-dimensional vector space over a field of characteristic zero and

ak{g} a subalgebra of ak{gl}(V) . Then V is semisimple as a module over ak{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).

References

• Book: Jacobson, Nathan . Lie algebras . 1979 . 1962 . Dover Publications, Inc. . New York . 978-0-486-13679-0 . 867771145 .