Triple

T6282520
Position Surface form Disambiguated ID Type / Status
Subject Lie algebra representation E140812 entity
Predicate relatedConcept P37 FINISHED
Object Verma module
A Verma module is a type of highest-weight module over a Lie algebra that is freely generated from a highest-weight vector and plays a central role in the classification of representations of semisimple Lie algebras.
E581261 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: Verma module | Statement: [Lie algebra representation, relatedConcept, Verma module]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Verma module
Context triple: [Lie algebra representation, relatedConcept, Verma module]
  • A. Harish-Chandra isomorphism
    The Harish-Chandra isomorphism is a fundamental result in representation theory that identifies the center of the universal enveloping algebra of a semisimple Lie algebra with the algebra of Weyl group–invariant polynomials on a Cartan subalgebra.
  • B. Harish-Chandra character formula
    The Harish-Chandra character formula is a fundamental result in representation theory that gives an explicit expression for the characters of irreducible admissible representations of real reductive Lie groups.
  • C. Weyl character formula
    The Weyl character formula is a fundamental result in representation theory that gives an explicit expression for the characters of irreducible finite-dimensional representations of semisimple Lie algebras and Lie groups.
  • D. Hecke operators
    Hecke operators are algebraic operators acting on modular forms that play a central role in number theory, particularly in understanding congruences, L-functions, and the arithmetic of modular forms.
  • E. Gelfand–Tsetlin basis
    The Gelfand–Tsetlin basis is a canonical, combinatorially defined basis for representations of certain Lie algebras and groups, particularly used in the representation theory of GL(n) and related structures.
  • 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: Verma module
Triple: [Lie algebra representation, relatedConcept, Verma module]
Generated description
A Verma module is a type of highest-weight module over a Lie algebra that is freely generated from a highest-weight vector and plays a central role in the classification of representations of semisimple Lie algebras.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Verma module
Target entity description: A Verma module is a type of highest-weight module over a Lie algebra that is freely generated from a highest-weight vector and plays a central role in the classification of representations of semisimple Lie algebras.
  • A. Harish-Chandra isomorphism
    The Harish-Chandra isomorphism is a fundamental result in representation theory that identifies the center of the universal enveloping algebra of a semisimple Lie algebra with the algebra of Weyl group–invariant polynomials on a Cartan subalgebra.
  • B. Harish-Chandra character formula
    The Harish-Chandra character formula is a fundamental result in representation theory that gives an explicit expression for the characters of irreducible admissible representations of real reductive Lie groups.
  • C. Weyl character formula
    The Weyl character formula is a fundamental result in representation theory that gives an explicit expression for the characters of irreducible finite-dimensional representations of semisimple Lie algebras and Lie groups.
  • D. Hecke operators
    Hecke operators are algebraic operators acting on modular forms that play a central role in number theory, particularly in understanding congruences, L-functions, and the arithmetic of modular forms.
  • E. Gelfand–Tsetlin basis
    The Gelfand–Tsetlin basis is a canonical, combinatorially defined basis for representations of certain Lie algebras and groups, particularly used in the representation theory of GL(n) and related structures.
  • 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_69c008cd17c8819082b82d3fbeb68047 completed March 22, 2026, 3:20 p.m.
NER Named-entity recognition batch_69c063f956c08190ae0f198ccbd68b42 completed March 22, 2026, 9:49 p.m.
NED1 Entity disambiguation (via context triple) batch_69c51962132881909a2eccd1203e03c1 completed March 26, 2026, 11:32 a.m.
NEDg Description generation batch_69c51b4803e08190ac067896da3400e5 completed March 26, 2026, 11:40 a.m.
NED2 Entity disambiguation (via description) batch_69c51bf81cfc8190a6f0e4ca74c7ff05 completed March 26, 2026, 11:43 a.m.
Created at: March 22, 2026, 4:26 p.m.