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Bibliographic record
Abstract
The classical Cayley map, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X right-arrow from bar left-parenthesis upper I Subscript n Baseline minus upper X right-parenthesis left-parenthesis upper I Subscript n Baseline plus upper X right-parenthesis Superscript negative 1"> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo stretchy="false"> ↦ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>I</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo> − </mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>I</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:mi>X</mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">X \mapsto (I_n-X)(I_n+X)^{-1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , is a birational isomorphism between the special orthogonal group <bold>SO</bold> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Subscript n"> <mml:semantics> <mml:msub> <mml:mi/> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and its Lie algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German s o Subscript n"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">s</mml:mi> </mml:mrow> <mml:mi>o</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">{\mathfrak so}_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , which is <bold>SO</bold> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Subscript n"> <mml:semantics> <mml:msub> <mml:mi/> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -equivariant with respect to the conjugating and adjoint actions, respectively. We ask whether or not maps with these properties can be constructed for other algebraic groups. We show that the answer is usually “no", with a few exceptions. In particular, we show that a Cayley map for the group <bold>SL</bold> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Subscript n"> <mml:semantics> <mml:msub> <mml:mi/> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> exists if and only if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n less-than-or-slanted-equals 3"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo> ⩽ </mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n \leqslant 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , answering an old question of <sc>Luna</sc> .
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.
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 | 0.002 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| 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