Type 𝐼𝐼 quantum subgroups of 𝔰𝔩_{𝔑}. ℑ: Symmetries of local modules
Bibliographic record
Abstract
This paper is the first of a pair that aims to classify a large number of the type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I upper I"> <mml:semantics> <mml:mrow> <mml:mi>I</mml:mi> <mml:mi>I</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">II</mml:annotation> </mml:semantics> </mml:math> </inline-formula> quantum subgroups of the categories <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C left-parenthesis German s German l Subscript r plus 1 Baseline comma k right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">s</mml:mi> <mml:mi mathvariant="fraktur">l</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>r</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:mi>k</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {C}(\mathfrak {sl}_{r+1}, k)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In this work we classify the braided auto-equivalences of the categories of local modules for all known type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I"> <mml:semantics> <mml:mi>I</mml:mi> <mml:annotation encoding="application/x-tex">I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> quantum subgroups of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C left-parenthesis German s German l Subscript r plus 1 Baseline comma k right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">s</mml:mi> <mml:mi mathvariant="fraktur">l</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>r</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:mi>k</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {C}(\mathfrak {sl}_{r+1}, k)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We find that the symmetries are all non-exceptional except for four cases (up to level-rank duality). These exceptional cases are the orbifolds <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C left-parenthesis German s German l Subscript 2 Baseline comma 16 right-parenthesis Subscript upper R e p left-parenthesis double-struck upper Z 2 right-parenthesis Superscript 0"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">s</mml:mi> <mml:mi mathvariant="fraktur">l</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:mn>16</mml:mn> <mml:msubsup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>Rep</mml:mi> <mml:mo> </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mn>0</mml:mn> </mml:msubsup> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {C}(\mathfrak {sl}_{2}, 16)^0_{\operatorname {Rep}(\mathbb {Z}_{2})}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C left-parenthesis German s German l Subscript 3 Baseline comma 9 right-parenthesis Subscript upper R e p left-parenthesis double-struck upper Z 3 right-parenthesis Superscript 0"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">s</mml:mi> <mml:mi mathvariant="fraktur">l</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>3</mml:mn> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:mn>9</mml:mn> <mml:msubsup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>Rep</mml:mi> <mml:mo> </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>3</mml:mn> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mn>0</mml:mn> </mml:msubsup> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.003 |
| Science and technology studies | 0.000 | 0.006 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.005 | 0.003 |
| 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".