nLab
reductive Lie algebra

Definition

A Lie algebra $𝔤$ is called reductive if the following equivalence conditions hold:

1. it is the direct sum $𝔤\simeq 𝔥\oplus 𝔞$ of a semisimple Lie algebra $𝔥$ and an abelian Lie algebra $𝔞$;

2. its adjoint representation is completely reducible?: every invariant subspace has an invariant complement.

Properties

• The Lie algebra of a compact and connected Lie group is reductive. (GHV, vol III, section 4.4.).

• The graded algebra of invariant polynomials on a reductive Lie algebra is the free graded algebra on the graded vector space of indecomposable invariant polynomials, and via transgression there generators are in bijection with the odd generators of the Lie algebra cohomology.

References

For instance volume III of

• Werner Greub, Stephen Halperin, Ray Vanstone, Connections, Curvature, and Cohomology Academic Press (1973)