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.

Two-dimensional highest-weight module data for gl(1|1)

Definition

For $a, b \in C$ , let $K (a, b) = C v \oplus C w$ , with $v$ even and $w$ odd. Specify the candidate action of the basis in Bracket data for gl(1|1) by $h_{1} v = a v, h_{2} v = b v, h_{1} w = (a - 1) w, h_{2} w = (b + 1) w, e v = 0, e w = (a + b) v, f v = w, f w = 0.$ The verification item The two-dimensional gl(1|1) action is well-defined ↗ proves that this data makes $K (a, b)$ a $g l (1 ∣ 1)$ -supermodule. Relative to the triangular decomposition in The gl(1|1) bracket data define a Lie superalgebra, define the algebraic-dual functional (in the sense of Linear functionals and the algebraic dual $V^{*} = L (V, F)$ ) $λ_{a, b} (c_{1} h_{1} + c_{2} h_{2}) = c_{1} a + c_{2} b$ on $h = C h_{1} \oplus C h_{2}$ ; then $v$ is a highest-weight vector of weight $λ_{a, b}$ .

Depends on

Used by

Finite-dimensional Lie supermodule categories need not be semisimple Counterexample
Typical and atypical highest weights for gl(1|1) Definition
The typicality boundary in a gl(1|1) highest-weight module Example
The two-dimensional gl(1|1) action is well-defined Theorem

Dependency tree · two levels

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Sources

Sigiswald Barbier, A minimal representation for the orthosymplectic Lie-superalgebra, §§2.4 and 2.6 (standard reference, not scraped)
Vera Serganova, Kac–Moody Superalgebras and Integrability, §§10 and 13 (standard reference, not scraped)