The Structure of H-(co)module Lie algebras
Bibliographic record
Abstract
Let L be a finite dimensional Lie algebra over a field of characteristic 0 .Then by the original Levi theorem, L = B R where R is the solvable radical and B is some maximal semisimple subalgebra.We prove that if L is an H -(co)module algebra for a finite dimensional (co)semisimple Hopf algebra H , then R is H -(co)invariant and B can be chosen to be H -(co)invariant too.Moreover, the nilpotent radical N of L is H -(co)invariant and there exists an H -sub(co)module S R such that R = S N and [B, S] = 0 .In addition, the H -(co)invariant analog of the Weyl theorem is proved.In fact, under certain conditions, these results hold for an H -comodule Lie algebra L , even if H is infinite dimensional.In particular, if L is a Lie algebra graded by an arbitrary group G, then B can be chosen to be graded, and if L is a Lie algebra with a rational action of a reductive affine algebraic group G by automorphisms, then B can be chosen to be G-invariant.Also we prove that every finite dimensional semisimple H -(co)module Lie algebra over a field of characteristic 0 is a direct sum of its minimal H -(co)invariant ideals.
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| 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".