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

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Upper central series of a Lie algebra

Definition

The upper central series of a Lie algebra $g$ is the ascending sequence of ideals

$Z_{0} (g) = 0, Z_{r + 1} (g) / Z_{r} (g) = Z (g / Z_{r} (g)) .$

Here the right side is the center (Lie subalgebras, ideals, and center) of the quotient Lie algebra (Quotient Lie algebras). Equivalently,

$Z_{r + 1} (g) = {x \in g : [g, x] \subseteq Z_{r} (g)} .$

The equivalence also shows inductively that $Z_{r + 1}$ is an ideal containing $Z_{r}$ : it is the inverse image of a central ideal under the quotient map. In particular $Z_{1} (g) = Z (g)$ . For the zero Lie algebra every term is zero.

Depends on

Used by

Nilpotent adjoint action yields a central series Corollary
Lower and upper central series characterize nilpotence Theorem

Dependency tree · two levels

5 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

Milne, Lie Algebras, Proposition 2.5 (standard reference, not scraped)