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reductive Lie algebra

Context

\infty-Lie theory

∞-Lie theory (higher geometry)

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.

More generally:

Definition

(Lie algebra reductive in ambient Lie algebra)

A sub-Lie algebra

𝔥𝔤 \mathfrak{h} \hookrightarrow \mathfrak{g}

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

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

Properties

References

Last revised on February 26, 2018 at 14:50:24. See the history of this page for a list of all contributions to it.