Triple

T21953427
Position Surface form Disambiguated ID Type / Status
Subject Lie algebra E542122 entity
Predicate hasKeyResult P70725 FINISHED
Object Weyl's theorem on complete reducibility
Weyl's theorem on complete reducibility is a fundamental result in the representation theory of semisimple Lie algebras stating that every finite-dimensional representation decomposes into a direct sum of irreducible subrepresentations.
E1510870 NE FINISHED

How this triple was built (4 steps)

Every LLM step that produced this triple, in pipeline order — named-entity classification, the disambiguation choices (the exact options shown, with the pick highlighted), and the generated description. The batch + timestamp of each is in the Provenance table below.

NER Named-entity recognition gpt-5-mini
Instruction
Given a phrase, classify it is english named entity (e.g., persons, organizations, works of art) in Latin script, or not (e.g., literals, dates, URLs, verbose phrases). For disambiguation, the statement where the phrase occurs as object is also given. Please return a JSON object with `phrase` (string, the phrase being analyzed) and `is_ne` (boolean, indicating whether the phrase is a Named Entity).
Input
Phrase: Weyl's theorem on complete reducibility | Statement: [Lie algebra, hasKeyResult, Weyl's theorem on complete reducibility]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Weyl's theorem on complete reducibility
Context triple: [Lie algebra, hasKeyResult, Weyl's theorem on complete reducibility]
  • A. Peter–Weyl theorem
    The Peter–Weyl theorem is a fundamental result in representation theory and harmonic analysis that decomposes square-integrable functions on a compact topological group into a direct sum of finite-dimensional irreducible unitary representations.
  • B. Harish-Chandra regularity theorem
    The Harish-Chandra regularity theorem is a fundamental result in representation theory that asserts characters of irreducible admissible representations of real reductive Lie groups are given by real-analytic, locally integrable functions on the group.
  • C. Borel–Weil theorem
    The Borel–Weil theorem is a fundamental result in representation theory that realizes irreducible representations of compact Lie groups as spaces of holomorphic sections of line bundles over their flag manifolds.
  • D. Schur’s lemma
    Schur’s lemma is a fundamental result in representation theory stating that any homomorphism between irreducible representations is either zero or an isomorphism, and that endomorphisms of an irreducible representation over an algebraically closed field are scalar multiples of the identity.
  • E. Maschke’s theorem
    Maschke’s theorem is a fundamental result in representation theory stating that every finite group representation over a field of characteristic not dividing the group order is completely reducible into a direct sum of irreducible representations.
  • F. None of above. chosen
  • G. Unsure - the case is ambiguous/there is not enough information to decide.
NEDg Description generation gpt-5.1
Instruction
Generate a one-sentence description of the target entity. 
You are given a context triple in the form (subject, predicate, object), where the object is the target entity. 
# Instructions
Use the triple to infer relevant information about the entity. Describe the entity based on what is most defining, well-known. 
Avoid repeating the information from the triple, unless really essential.
# Response Format
Return only the sentence: "Description: [one-sentence description of the target entity]"
Input
Entity: Weyl's theorem on complete reducibility
Triple: [Lie algebra, hasKeyResult, Weyl's theorem on complete reducibility]
Generated description
Weyl's theorem on complete reducibility is a fundamental result in the representation theory of semisimple Lie algebras stating that every finite-dimensional representation decomposes into a direct sum of irreducible subrepresentations.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Weyl's theorem on complete reducibility
Target entity description: Weyl's theorem on complete reducibility is a fundamental result in the representation theory of semisimple Lie algebras stating that every finite-dimensional representation decomposes into a direct sum of irreducible subrepresentations.
  • A. Peter–Weyl theorem
    The Peter–Weyl theorem is a fundamental result in representation theory and harmonic analysis that decomposes square-integrable functions on a compact topological group into a direct sum of finite-dimensional irreducible unitary representations.
  • B. Harish-Chandra regularity theorem
    The Harish-Chandra regularity theorem is a fundamental result in representation theory that asserts characters of irreducible admissible representations of real reductive Lie groups are given by real-analytic, locally integrable functions on the group.
  • C. Borel–Weil theorem
    The Borel–Weil theorem is a fundamental result in representation theory that realizes irreducible representations of compact Lie groups as spaces of holomorphic sections of line bundles over their flag manifolds.
  • D. Schur’s lemma
    Schur’s lemma is a fundamental result in representation theory stating that any homomorphism between irreducible representations is either zero or an isomorphism, and that endomorphisms of an irreducible representation over an algebraically closed field are scalar multiples of the identity.
  • E. Maschke’s theorem
    Maschke’s theorem is a fundamental result in representation theory stating that every finite group representation over a field of characteristic not dividing the group order is completely reducible into a direct sum of irreducible representations.
  • F. None of above. chosen

Provenance (5 batches)

The batch behind each pipeline step, in order, with when it ran. Timestamps are batch-level — stages were processed in waves, so the object chain (NER → NED1 → NEDg → NED2) reads in order, but predicate / elicitation batches can sit in a different wave.

Step Stage Batch ID Status When
creating Elicitation batch_69e0c47ef0e48190a50e1bcc43f4b3fd completed April 16, 2026, 11:14 a.m.
NER Named-entity recognition batch_69f1243dfb4081909bc7a722843ffea7 completed April 28, 2026, 9:18 p.m.
NED1 Entity disambiguation (via context triple) batch_6a0a670672b481908b92e15245212e64 completed May 18, 2026, 1:10 a.m.
NEDg Description generation batch_6a0a67c29848819080860f13b4c9697b completed May 18, 2026, 1:13 a.m.
NED2 Entity disambiguation (via description) batch_6a0a686a0f208190977890119ffcff79 completed May 18, 2026, 1:16 a.m.
Created at: April 16, 2026, 7:59 p.m.