DefinitionDefinition: AI-adaptedProof: 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.

Typical and atypical highest weights for gl(1|1)

Definition

For the fixed $g l (1 ∣ 1)$ bracket and positive odd root $e$ of The gl(1|1) bracket data define a Lie superalgebra, let $λ_{a, b}$ be the highest weight with $λ_{a, b} (h_{1}) = a$ and $λ_{a, b} (h_{2}) = b$ . For the explicit module family $K (a, b)$ of Two-dimensional highest-weight module data for gl(1|1), call this parameter typical when $a + b \neq 0$ and atypical when $a + b = 0$ . This is the rank-one $g l (1 ∣ 1)$ convention for this page only; no typicality criterion for another Lie superalgebra or root datum is asserted.

Depends on

Used by

The typicality boundary in a gl(1|1) highest-weight module Example

Dependency tree · two levels

9 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, §13 (standard reference, not scraped)
Sigiswald Barbier, A minimal representation for the orthosymplectic Lie-superalgebra, §2.6 (standard reference, not scraped)