Reductive Lie algebra

Summary

A reductive Lie algebra is a finite-dimensional Lie algebra whose adjoint representation is completely reducible. In characteristic $0$ this means that it is the direct sum of an abelian Lie algebra and a semisimple Lie algebra. The main examples are $\mathfrak{gl}_n$ and the Lie algebras of compact Lie groups. Reductive Lie algebras are the natural setting for much of representation theory, since $\mathfrak{gl}_n$ is reductive but not semisimple.