DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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Quotient Lie algebras

Definition

For an ideal $i ⊴ g$ , the quotient vector space $g / i$ equipped with

$[x + i, y + i] = [x, y] + i$

is the quotient Lie algebra. The bracket is representative-independent and satisfies all Lie identities by The quotient Lie-algebra bracket is well-defined. The canonical projection $π : g \to g / i$ is linear and satisfies $π ([x, y]) = [π (x), π (y)]$ .

Depends on

The quotient Lie-algebra bracket is well-defined

Used by

Codimension-one ideals in nilpotent Lie algebras Corollary
A nilpotent-by-nilpotent extension need not be nilpotent Counterexample
Lie algebra presented by generators and relations Definition
Upper central series of a Lie algebra Definition
First cohomology with trivial coefficients Example
Nilpotent-by-nilpotent extensions are always nilpotent False statement
Codimension-one ideal in a nonzero solvable Lie algebra Lemma
A central extension of a nilpotent Lie algebra is nilpotent Proposition
Ideals and quotients of semisimple Lie algebras Proposition
Kernels, images, and the first isomorphism theorem for Lie algebras Proposition
Subalgebras, quotients, and extensions of solvable Lie algebras Proposition
Subalgebras, quotients, and finite products of nilpotent Lie algebras Proposition
The normalizer acts on Lie algebra cohomology Proposition
The radical is characteristic and its quotient has zero radical Proposition
Lower and upper central series characterize nilpotence Theorem

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Kirillov, An Introduction to Lie Groups and Lie Algebras, §5.3, before Lemma 5.17 (standard reference, not scraped)