Simple modules over the Lie algebras of divergence zero vector fields on a torus
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Bibliographic record
Abstract
Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> {n\geq 2} be an integer, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>𝕊</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> {\mathbb{S}_{n}} the Lie algebra of divergence zero vector fields on an n -dimensional torus, and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>𝒦</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> {\mathcal{K}_{n}} the Weyl algebra over the Laurent polynomial algebra <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>A</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo>=</m:mo> <m:mrow> <m:mi>ℂ</m:mi> <m:mo></m:mo> <m:mrow> <m:mo>[</m:mo> <m:msubsup> <m:mi>x</m:mi> <m:mn>1</m:mn> <m:mrow> <m:mo>±</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msubsup> <m:mo>,</m:mo> <m:msubsup> <m:mi>x</m:mi> <m:mn>2</m:mn> <m:mrow> <m:mo>±</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msubsup> <m:mo>,</m:mo> <m:mi>…</m:mi> <m:mo>,</m:mo> <m:msubsup> <m:mi>x</m:mi> <m:mi>n</m:mi> <m:mrow> <m:mo>±</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msubsup> <m:mo>]</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {A_{n}=\mathbb{C}[x_{1}^{\pm 1},x_{2}^{\pm 1},\dots,x_{n}^{\pm 1}]} . For any <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>𝔰</m:mi> <m:mo></m:mo> <m:msub> <m:mi>𝔩</m:mi> <m:mi>n</m:mi> </m:msub> </m:mrow> </m:math> {\mathfrak{sl}_{n}} -module V and any module P over <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>𝒦</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> {\mathcal{K}_{n}} , we define an <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>𝕊</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> {\mathbb{S}_{n}} -module structure on the tensor product <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>P</m:mi> <m:mo>⊗</m:mo> <m:mi>V</m:mi> </m:mrow> </m:math> {P\otimes V} . In this paper, necessary and sufficient conditions for the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>𝕊</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> {\mathbb{S}_{n}} -modules <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>P</m:mi> <m:mo>⊗</m:mo> <m:mi>V</m:mi> </m:mrow> </m:math> {P\otimes V} </jats:alternat
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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 | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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 it