Bibliographic record
Abstract
We consider Hilbert series of ordinary Lie algebras, restricted (or p-) Lie algebras, and color Lie (p-)superalgebras. We derive a dimension formula similar to a wellknown Witt’s formula for free color Lie superalgebras and a certain class of color Lie p-superalgebras. A Lie (super)algebra analogue of a well-known Schreier’s formula for the rank of a subgroup of finite index in a free group was found by V. M. Petrogradsky. In this dissertation, Petrogradsky’s formulas are extended to the case of color Lie (p-)superalgebras. We establish more Schreier-type formulas for the ranks of submodules of free modules over free associative algebras and free group algebras. As an application, we consider Hopf subalgebras of some cocommutative Hopf algebras. Also, we apply our version of Witt and Schreier formulas to study relatively free color Lie (p-)superalgebras and to prove that the free color Lie superalgebra and its enveloping algebra have the same entropy. Y. A. Bahturin and A. Y. Olshanskii proved that the relative growth rate of a finitely generated subalgebra K of a free Lie algebra L of finite rank is strictly less than the growth rate of the free Lie algebra itself. We show that this theorem cannot be extended to free color Lie superalgebras in general. However, we establish it in a special case.
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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 | 0.002 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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".