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Bibliographic record
Abstract
We introduce and study the quantum version of the differential operator algebra on Laurent polynomials and its associated Lie algebra over a field F of characteristic 0. The q-quantum torus F q is the unital associative algebra over F generated by [Formula: see text] subject to the defining relations t i t j = q i,j t j t i , where q i,i = 1, [Formula: see text]. Let D be a subspace of [Formula: see text] where ∂ i is the derivation on F q sending [Formula: see text] to [Formula: see text]. Then, the quantum differential operator algebra is the associative algebra F q [D]. Assume that F q [D] is simple as an associative algebra. We compute explicitly all 2-cocycles of F q [D], viewed as a Lie algebra. More precisely, we show that the second cohomology group of F q [D] has dimension n if D = 0, dimension 1 if dim D = 1, and dimension 0 if dim D > 1. We also determine all isomorphisms and anti-isomorphisms F q [D] → F q′ [D′] of simple associative algebras, and all isomorphisms F q [D]/F → F q′ [D′]/F of simple Lie algebras.
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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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| 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 it