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.

Simple and semisimple Lie supermodules

Definition

A graded submodule of a Lie supermodule $V = V_{\overset{ˉ}{0}} \oplus V_{\overset{ˉ}{1}}$ is a subspace $W = W_{\overset{ˉ}{0}} \oplus W_{\overset{ˉ}{1}}$ , with $W_{\overset{ˉ}{i}} \subseteq V_{\overset{ˉ}{i}}$ , invariant under the $g$ -action. A nonzero supermodule is simple if its only graded submodules are $0$ and itself. The zero supermodule is not simple.

A supermodule is semisimple if it is an internal direct sum of simple graded submodules; the zero module is the empty direct sum and is semisimple. The category of finite-dimensional Lie supermodules for $g$ is semisimple if every finite-dimensional object is semisimple. These definitions are in the graded category, whose morphisms preserve parity.

Depends on

Lie supermodules and representations

Used by

Finite-dimensional Lie supermodule categories need not be semisimple Counterexample
The typicality boundary in a gl(1|1) highest-weight module Example

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

Sigiswald Barbier, A minimal representation for the orthosymplectic Lie-superalgebra, §2.4.1 (standard reference, not scraped)
Vera Serganova, Kac–Moody Superalgebras and Integrability, §10 (standard reference, not scraped)