Reductive Lie algebra

A Lie algebra $\mathfrak{g}$ is reductive if it can be written as

where $Z(\mathfrak{g})$ is the center and $[\mathfrak{g}, \mathfrak{g}]$ is the derived algebra.

Equivalent characterizations

The following are equivalent for a finite-dimensional Lie algebra over a field of characteristic zero:

1. $\mathfrak{g}$ is reductive.
2. The adjoint representation is completely reducible.
3. $\mathfrak{g}$ is a direct sum of simple and abelian Lie algebras.
4. Every finite-dimensional representation is completely reducible.

Examples

• Semisimple Lie algebras (center is trivial).
• Abelian Lie algebras (derived algebra is trivial).
• $\mathfrak{gl}_n$: center is scalar matrices, derived algebra is $\mathfrak{sl}_n$.
• $\mathfrak{u}(n) = i\mathfrak{u}(1) \oplus \mathfrak{su}(n)$.

Non-example

The Lie algebra of upper triangular matrices is not reductive (it is solvable but not semisimple).