Meromorphic tensor equivalence for Yangians and quantum loop algebras
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Abstract
Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="fraktur">g</mml:mi> </mml:math> be a complex semisimple Lie algebra, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>Y</mml:mi> <mml:mi>ħ</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="fraktur">g</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>q</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>L</mml:mi> <mml:mi mathvariant="fraktur">g</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> the corresponding Yangian and quantum loop algebra, with deformation parameters related by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>q</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow> <mml:mi>π</mml:mi> <mml:mi>ι</mml:mi> <mml:mi>ħ</mml:mi> </mml:mrow> </mml:msup> </mml:math> . When <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>ħ</mml:mi> </mml:math> is not a rational number, we constructed in Gautam and Toledano Laredo (J. Am. Math. Soc. 29:775, 2016) a faithful functor <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="normal">Γ</mml:mi> </mml:math> from the category of finite-dimensional representations of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>Y</mml:mi> <mml:mi>ħ</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="fraktur">g</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> to those of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>q</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>L</mml:mi> <mml:mi mathvariant="fraktur">g</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> . The functor <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="normal">Γ</mml:mi> </mml:math> is governed by the additive difference equations defined by the commuting fields of the Yangian, and restricts to an equivalence on a subcategory of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mo>Rep</mml:mo> <mml:mo>fd</mml:mo> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>Y</mml:mi> <mml:mi>ħ</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="fraktur">g</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:math> defined by choosing a branch of the logarithm. In this paper, we construct a tensor structure on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="normal">Γ</mml:mi> </mml:math> and show that, if <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo stretchy="false">|</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy="false">|</mml:mo> <mml:mo>≠</mml:mo> <mml:mn>1</mml:mn> </mml:math> , it yields an equivalence of meromorphic braided tensor categories, when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>Y</mml:mi> <mml:mi>ħ</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="fraktur">g</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>q</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>L</mml:mi> <mml:mi mathvariant="fraktur">g</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> are endowed with the deformed Drinfeld coproducts and the commutative part of their universal <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>R</mml:mi> </mml:math> -matrices. This proves in particular the Kohno–Drinfeld theorem for the abelian <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>q</mml:mi> </mml:math> KZ equations defined by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>Y</mml:mi> <mml:mi>ħ</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="fraktur">g</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> . The tensor structure arises from the abelian <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>q</mml:mi> </mml:math> KZ equations defined by an appropriate regularisation of the commutative part of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>R</mml:mi> </mml:math> -matrix of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>Y</mml:mi> <mml:mi>ħ</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="fraktur">g</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> .
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.001 |
| Open science | 0.001 | 0.000 |
| 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 it