DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Triangular decomposition of a Lie superalgebra

Definition

A triangular decomposition datum for a complex Lie superalgebra $g$ is a triple of graded subalgebras $(n^{-}, h, n^{+})$ such that $h \subseteq g_{\overset{ˉ}{0}}$ is abelian, $g = n^{-} \oplus h \oplus n^{+}$ as graded vector spaces, and $b : = h \oplus n^{+}$ is a subalgebra. No root decomposition, nondegenerate form, basic-classical hypothesis, or finite-dimensionality is implicit in this definition.

The subalgebra $b$ is the selected positive subalgebra. Different choices of $n^{+}$ can give different highest-weight conventions for the same superalgebra.

Depends on

Lie superalgebra over the complex numbers

Used by

Highest-weight vectors and modules for a Lie superalgebra Definition
The gl(1|1) bracket data define a Lie superalgebra Theorem

Dependency tree · two levels

3 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

Vera Serganova, Kac–Moody Superalgebras and Integrability, §2 (standard reference, not scraped)
Sigiswald Barbier, A minimal representation for the orthosymplectic Lie-superalgebra, §2.5 (standard reference, not scraped)