Linear Lie algebra

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

References

• Jacobson, Nathan (1979) [1962]. Lie algebras. New York: Dover Publications, Inc. ISBN 978-0-486-13679-0. OCLC 867771145.