DEVELOPMENT OF A UNIFIED TENSOR CALCULUS FOR THE EXCEPTIONAL LIE ALGEBRAS
Bibliographic record
Abstract
The uniformity of the decomposition law, for a family ℱ of Lie algebras which includes the exceptional Lie algebras, of the tensor powers ad ⊗n of their adjoint representations ad is now well known. This paper uses it to embark on the development of a unified tensor calculus for the exceptional Lie algebras. It deals explicitly with all the tensors that arise at the n=2 stage, obtaining a large body of systematic information about their properties and identities satisfied by them. Some results at the n=3 level are obtained, including a simple derivation of the dimension and Casimir eigenvalue data for all the constituents of ad ⊗3 . This is vital input data for treating the set of all tensors that enter the picture at the n=3 level, following a path already known to be viable for [Formula: see text]. The special way in which the Lie algebra [Formula: see text] conforms to its place in the family ℱ alongside the exceptional Lie algebras is described.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 teacher head, 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".