Triple
T5705600
| Position | Surface form | Disambiguated ID | Type / Status |
|---|---|---|---|
| Subject | Cartan decomposition |
E125775
|
entity |
| Predicate | involves |
P1256
|
FINISHED |
| Object |
Cartan involution
A Cartan involution is a specific type of involutive automorphism of a Lie algebra or Lie group that enables the decomposition of the structure into compact and non-compact parts, playing a central role in the classification of semisimple Lie algebras.
|
E125775
|
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: Cartan involution | Statement: [Cartan decomposition, involves, Cartan involution]
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Cartan involution Context triple: [Cartan decomposition, involves, Cartan involution]
-
A.
Cartan decomposition
Cartan decomposition is a fundamental structural result in Lie theory that expresses a Lie algebra or Lie group as a direct sum or product of subspaces or subgroups with specific symmetry properties, widely used in differential geometry and representation theory.
-
B.
Cartan–Killing form
The Cartan–Killing form is a canonical symmetric bilinear form on a Lie algebra that plays a central role in classifying and studying the structure of Lie algebras and Lie groups.
-
C.
Cartan
Cartan is a French surname most famously associated with mathematician Élie Cartan and his influential family of mathematicians.
-
D.
Cartan subalgebras
Cartan subalgebras are maximal abelian subalgebras of a Lie algebra consisting of semisimple elements, fundamental for classifying and understanding the structure of Lie algebras.
-
E.
Cartan theorems A and B
Cartan theorems A and B are fundamental results in complex analytic geometry that characterize coherent analytic sheaves on Stein spaces by guaranteeing the existence of enough global sections and the vanishing of higher cohomology.
- 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: Cartan involution Triple: [Cartan decomposition, involves, Cartan involution]
Generated description
A Cartan involution is a specific type of involutive automorphism of a Lie algebra or Lie group that enables the decomposition of the structure into compact and non-compact parts, playing a central role in the classification of semisimple Lie algebras.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Cartan involution Target entity description: A Cartan involution is a specific type of involutive automorphism of a Lie algebra or Lie group that enables the decomposition of the structure into compact and non-compact parts, playing a central role in the classification of semisimple Lie algebras.
-
A.
Cartan decomposition
chosen
Cartan decomposition is a fundamental structural result in Lie theory that expresses a Lie algebra or Lie group as a direct sum or product of subspaces or subgroups with specific symmetry properties, widely used in differential geometry and representation theory.
-
B.
Cartan–Killing form
The Cartan–Killing form is a canonical symmetric bilinear form on a Lie algebra that plays a central role in classifying and studying the structure of Lie algebras and Lie groups.
-
C.
Cartan
Cartan is a French surname most famously associated with mathematician Élie Cartan and his influential family of mathematicians.
-
D.
Cartan subalgebras
Cartan subalgebras are maximal abelian subalgebras of a Lie algebra consisting of semisimple elements, fundamental for classifying and understanding the structure of Lie algebras.
-
E.
Cartan theorems A and B
Cartan theorems A and B are fundamental results in complex analytic geometry that characterize coherent analytic sheaves on Stein spaces by guaranteeing the existence of enough global sections and the vanishing of higher cohomology.
- F. None of above.
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_69c0082c96988190b3a6a201edce472a |
completed | March 22, 2026, 3:18 p.m. |
| NER | Named-entity recognition | batch_69c02459cd18819080fda0b481d11f08 |
completed | March 22, 2026, 5:18 p.m. |
| NED1 | Entity disambiguation (via context triple) | batch_69c07de7df8c8190824d24f729eaa04d |
completed | March 22, 2026, 11:40 p.m. |
| NEDg | Description generation | batch_69c08b820a048190b3874522d568d485 |
completed | March 23, 2026, 12:38 a.m. |
| NED2 | Entity disambiguation (via description) | batch_69c08be237a88190ace6e3d4ab97bf17 |
completed | March 23, 2026, 12:40 a.m. |
Created at: March 22, 2026, 3:45 p.m.