Bibliographic record
Abstract
This work is concerned with Mishchenko and Fomenko's celebrated theory of completely integrable systems on a complex semisimple Lie algebra \mathfrak{g} . Their theory associates a maximal Poisson-commutative subalgebra of \mathbb{C}[\mathfrak{g}] to each regular element a\in\mathfrak{g} , and one can assemble free generators of this subalgebra into a moment map F_a:\mathfrak{g}\rightarrow\mathbb{C}^b . This leads one to pose basic structural questions about F_a and its fibres, e.g. questions concerning the singular points and irreducible components of such fibres. We examine the structure of fibres in Mishchenko-Fomenko systems, building on the foundation laid by Bolsinov, Charbonnel-Moreau, Moreau, and others. This includes proving that the critical values of F_a have codimension 1 or 2 in \mathbb{C}^b , and that each codimension is achievable in examples. Our results on singularities make use of a subalgebra \mathfrak{b}^a\subseteq\mathfrak{g} , defined to be the intersection of all Borel subalgebras of \mathfrak{g} containing a . In the case of a non-nilpotent a\in\mathfrak{g}_{\mathrm{reg}} and an element x\in\mathfrak{b}^a , we prove the following: x+[\mathfrak{b}^a,\mathfrak{b}^a] lies in the singular locus of F_a^{-1}(F_a(x)) , and the fibres through points in \mathfrak{b}^a form a \text{rank}(\mathfrak{g}) -dimensional family of singular fibres. We next consider the irreducible components of our fibres, giving a systematic way to construct many components via Mishchenko-Fomenko systems on Levi subalgebras \mathfrak{l}\subseteq\mathfrak{g} . In addition, we obtain concrete results on irreducible components that do not arise from the aforementioned construction. Our final main result is a recursive formula for the number of irreducible components in F_a^{-1}(0) , and it generalizes a result of Charbonnel-Moreau. Illustrative examples are included at the end of this paper.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.000 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
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 source (direct Gemma or distilled Codex), 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".