Bibliographic record
Abstract
We propose a quantum analogue of a Tits-Kantor-Koecher algebra with a Jordan torus as an coordinated algebra by looking at the vertex operator construction over a Fock space. Quantum toroidal algebras were first introduced by Ginzburg, Kapranov and Vasserot [GKV] in the study of the Langlands reciprocity for algebraic surfaces. These algebras are quantized analogues for toroidal Lie algebras of Moody-Rao-Yokonuma [MRY]. Representations of quantum toroidal algebras have been studied by Varagnolo-Vasserot [VV], Saito-Takemura-Uglov [STU], Saito [S], Frenkel-Jing-Wang [FJW], Takemura-Uglov[TU], Gao-Jing [GJ1,2], and among others. The Tits-Kantor-Koecher (TKK) algebra was originally defined from Jordan algebra in constructing the finite dimensional simple Lie algebras of the exceptional types E6 and E7. It has also played an important role in the structure theory of newly developed extended affine Lie algebras. A TKK algebra in the extended affine Lie algebras of type A1 has been realized by gluing a Clifford module and a Heisenberg module in Tan’s paper [T]. This algebra appears as the core of extended affine Lie algebras of type A1[AABGP] and has been studied by Yoshii [Y]. In this note, we shall propose a quantum analogue of the above Tits-Kantor-Koecher algebra. Our motivation comes from the vertex operator construction as was done in [GJ1, GJ2]. We hope that the quantum TKK algebra will be useful in the study of quantum toroidal algebras. Like the quantum Kac-Moody algebra case [J2] our construction relies on an interesting combinatorial identity of Hall-Littlewood type [M]. It suggests that representations
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.006 | 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".