nLab
reductive Lie algebra

Context

\infty-Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Definition

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

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

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

Properties

References

For instance volume III of

Revised on June 30, 2011 18:48:15 by Urs Schreiber (131.211.239.52)