Quantum representation theory and Manin matrices I: The finite-dimensional case
Bibliographic record
Abstract
We construct a theory that describes a quantum (non-commutative) analogue of representations within the framework of “non-commutative linear geometry” set out in the work of Manin [ <italic>Quantum groups and noncommutative geometry</italic> , Univ. Montréal, Centre de Recherches Mathématiques, Montréal, QC, 1988]. For this purpose, the concept of an internal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="hom"> <mml:semantics> <mml:mi>hom</mml:mi> <mml:annotation encoding="application/x-tex">\hom</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -functor is generalized to the case of parameterized adjunctions, and we construct a general approach to representations of monoids for a symmetric monoidal category with a parameter subcategory. A quantum theory of representations is then obtained by applying this approach to the monoidal category of a certain class of graded algebras with the Manin product, where the parameterizing subcategory consists of connected finitely generated quadratic algebras. We formulate this theory in the language of Manin matrices. We also obtain quantum analogues of the direct sum and tensor product of representations. Finally, we give some examples of quantum representations.
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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 | 0.001 | 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) | 0.000 | 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".