nLab reductive Lie algebra

Definition

A Lie algebra $\mathfrak{g}$ is called reductive if the following equivalent conditions hold:

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

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

Over a field of characteristic zero, the following conditions on $\mathfrak{g}$ are also equivalent to $\mathfrak{g}$ being reductive:

1. the radical of $\mathfrak{g}$ is equal to the centre of $\mathfrak{g}$ (in general, the radical is only contained inside the centre);

2. $\mathfrak{g}$ is a direct sum of its centre with a semisimple ideal;

3. $\mathfrak{g}$ is a direct sum of prime ideals.

More generally:

Definition

(Lie algebra reductive in ambient Lie algebra)

A sub-Lie algebra

$$\mathfrak{h} \hookrightarrow \mathfrak{g}$$

is called reductive if the adjoint Lie algebra representation of $\mathfrak{h}$ on $\mathfrak{g}$ is reducible.

(Koszul 50, recalled in e.g. Solleveld 02, def. 2.27)

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.

Related concepts

• reductive dual pair

References

• Jean-Louis Koszul, Homologie et cohomologie des algèbres de Lie, Bull. Soc. Math. France 78 (1950), 65-127

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

• Maarten Solleveld, Lie algebra cohomology and Macdonald’s conjectures, 2002 (pdf)