Cohomologies, extensions and deformations of differential algebras of arbitrary weight
Bibliographic record
Abstract
As an algebraic structure underlying the differential calculus and differential equations, a differential algebra is an associative algebra equipped with a linear map satisfying the Leibniz rule.The subject has been studied for about a century and has become an important area of mathematics.In recent years the area has been expanded to the noncommutative associative and Lie algebra contexts and to the case when the defining operator identity has a weight in order to include difference operators.This paper provides a cohomology theory for differential algebras of arbitrary weight, via a uniform approach to cover both the zero weight case which is similar to the earlier study of differential Lie algebras, and the non-zero weight case which poses challenges.The cohomology of a differential algebra is related to the Hochschild cohomology by a type of long exact sequence for relative homology.As an application, abelian extensions of a differential algebra are classified by the second cohomology group.Furthermore, formal deformations of a differential algebra are characterized by the second cohomology group and the rigidity of a differential algebra is characterized by the vanishing of the second cohomology group.Contents 1 Introduction 1409 2 Cohomology of differential algebras 1411 3 Abelian extensions of differential algebras 1418 4 Formal deformations of differential algebras 1424
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.000 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".