The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>R</mml:mi> </mml:math> -matrix formalism for quantized enveloping algebras
Bibliographic record
Abstract
Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>ℏ</mml:mi> </mml:msub> <mml:mi>𝔤</mml:mi> </mml:mrow> </mml:math> denote the Drinfeld–Jimbo quantum group associated to a simple Lie algebra <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝔤</mml:mi> </mml:math> . We apply a modification of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>R</mml:mi> </mml:math> -matrix construction for quantum groups to the evaluation of the universal <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>R</mml:mi> </mml:math> -matrix of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>ℏ</mml:mi> </mml:msub> <mml:mi>𝔤</mml:mi> </mml:mrow> </mml:math> on any of its finite-dimensional representations. This produces a quantized enveloping algebra <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal">U</mml:mi> <mml:mi mathvariant="normal">R</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>𝔤</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> whose definition is given in terms of two generating matrices satisfying variants of the well-known <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>R</mml:mi> <mml:mi>L</mml:mi> <mml:mi>L</mml:mi> </mml:mrow> </mml:math> relations. We prove that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal">U</mml:mi> <mml:mi mathvariant="normal">R</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>𝔤</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> is isomorphic to the tensor product of the quantum double of the Borel subalgebra <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>ℏ</mml:mi> </mml:msub> <mml:mi>𝔟</mml:mi> <mml:mo>⊂</mml:mo> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>ℏ</mml:mi> </mml:msub> <mml:mi>𝔤</mml:mi> </mml:mrow> </mml:math> and a quantized polynomial algebra encoded by the space of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝔤</mml:mi> </mml:math> -invariants associated to the semiclassical limit <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> of the underlying finite-dimensional representation. Using this description, we characterize <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>ℏ</mml:mi> </mml:msub> <mml:mi>𝔤</mml:mi> </mml:mrow> </mml:math> and the quantum double of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>ℏ</mml:mi> </mml:msub> <mml:mi>𝔟</mml:mi> </mml:mrow> </mml:math> as Hopf quotients of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal">U</mml:mi> <mml:mi mathvariant="normal">R</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>𝔤</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> and as fixed-point subalgebras with respect to certain natural automorphisms. As an additional corollary, we deduce that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal">U</mml:mi> <mml:mi mathvariant="normal">R</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>𝔤</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> is quasitriangular precisely when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> is multiplicity free.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.002 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 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 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".