Contractions and deformations of Lie algebras in Physics
Bibliographic record
Abstract
In this contribution, we discuss the mutually opposite procedures of deformations and contractions of Lie algebras. From the group‐theoretical point of view, Einstein’s special theory of relativity is predicated on the replacement of the Galilei group by one of its deformations: the Poincaré group. Conversely, the Galilei group can be obtained by a contraction of the Poincaré group. Our principal objective is to demonstrate that appropriate combinations of both procedures may lead to new Lie algebras and thereby to new physical theories as well as new insights. To illustrate this, we recall that infinite‐dimensional Lie algebras of Krichever‐Novikov type can be retrieved by deforming the Virasoro or Kac‐Moody algebras. Then, in turn, contrations of Krichever‐Novikov algebras lead to new infinite‐dimensional Lie algebras. We end by discussing the real three‐dimensional Lie algebras. We observe that, whereas for every contraction there exists a reverse deformation, the converse is not true in general.
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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 | 0.002 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.007 |
| Scholarly communication | 0.001 | 0.005 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 0.001 |
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".