Linear Lie algebra

In algebra, a linear Lie algebra is a subalgebra ${\mathfrak {g}}$ of the Lie algebra ${\mathfrak {gl}}(V)$ 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 ${\mathfrak {g}}$ (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 ${\mathfrak {gl}}(V)$. 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.