Representation Theory · arXiv:2010.00646 v2 · submitted 1 Oct 2020
$\imath$Hall algebra of the projective line and q-Onsager algebra
The paper says
“This $\imath$Hall algebra is shown to realize the universal q-Onsager algebra (i.e., $\imath$quantum group of split affine A1 type) in its Drinfeld type presentation.”
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Abstract
The $\imath$Hall algebra of the projective line is by definition the twisted semi-derived Ringel-Hall algebra of the category of 1-periodic complexes of coherent sheaves on the projective line. This $\imath$Hall algebra is shown to realize the universal q-Onsager algebra (i.e., $\imath$quantum group of split affine A1 type) in its Drinfeld type presentation. The $\imath$Hall algebra of the Kronecker quiver was known earlier to realize the same algebra in its Serre type presentation. We then establish a derived equivalence which induces an isomorphism of these two $\imath$Hall algebras, explaining the isomorphism of the q-Onsager algebra under the two presentations.
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abstract compiled · 2026-09-23 · no language-model review
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The abstract, sentence by sentence
This table labels the abstract only (arXiv listing abstract v2). The number is the probability the classifier gave the label.
| Sentence | Role | The authors' own work | States the main conclusion | Strength |
|---|---|---|---|---|
| The $\imath$Hall algebra of the projective line is by definition the twisted semi-derived Ringel-Hall algebra of the category of $1$-periodic complexes of coherent sheaves on the projective line. | background 0.91 | 0.12 | 0.05 | asserted 1.00 |
| This $\imath$Hall algebra is shown to realize the universal $q$-Onsager algebra (i.e., $\imath$quantum group of split affine $A_1$ type) in its Drinfeld type presentation. | result 0.97 | 0.92 | 0.79 | asserted 1.00 |
| The $\imath$Hall algebra of the Kronecker quiver was known earlier to realize the same algebra in its Serre type presentation. | background 1.00 | 0.04 | 0.05 | asserted 1.00 |
| We then establish a derived equivalence which induces an isomorphism of these two $\imath$Hall algebras, explaining the isomorphism of the $q$-Onsager algebra under the two presentations. | contribution 0.43 | 0.94 | 0.23 | asserted 1.00 |
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,758 input tokens.
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Reviewed papers on Pith that cite this one. The full citation record
How the literature uses it
8 arXiv papers cite this in 20 sentences
Papers citing it · 9 sentences without a confident label
Builds on it 1 paper
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This is a sequel of [ref] in our program devoted to developing a geometric realization of the $$quantum groups in Drinfeld type presentations, a vast generalization of [ref] .
$\imath$Hall algebras of weighted projective lines and quantum symmetric pairs · Goal · 0.98
Uses its method or tool 2 papers
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For any $K^∙∈_1,ac()$ and $M^∙∈_1()$, by [ref] , define [equation] These formulas give rise to well-defined bilinear forms (called Euler forms), again denoted by $⟨·, ·⟩$, on the Grothendieck groups $K_0(_1,ac())$ and $K_0(_1())$.
$\imath$quantum groups of split type via derived Hall algebras · The category of $1$-periodic complexes · 0.85
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Using the proof of [ref] , one can obtain the following remark.
$\imath$Hall algebras of weighted projective lines and quantum symmetric pairs · A derived equivalence · 0.85
Cites it as background 3 papers
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A Drinfeld type presentation of $O_q$ is obtained in [ref] , and this is used in [ref] to realize $O_q$ as an $ι$Hall algebra of the projective line.
The compact presentation for the alternating central extension of the $q$-Onsager algebra · Introduction · 0.86
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A Drinfeld type presentation of $O_q$ is obtained in [ref] , and this is used in [ref] to realize $O_q$ as an $ι$Hall algebra of the projective line.
The $S_3$-symmetric $q$-Onsager algebra and its Lusztig automorphisms · Introduction · 0.84
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Up to the present, $O_q$ remains an active research topic in mathematics [ref] and physics [ref] .
The alternating central extension of the Onsager Lie algebra · Introduction · 0.92
What it cites · 10 sentences without a confident label
Uses the method or tool of 1 paper
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Following [ref] , we consider the ideal $$ of $(_1())$ generated by
Hall algebras and quantum symmetric pairs of Kac-Moody type · $$Hall algebras · 0.81
Compares with 1 paper
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For any $x∈$ and $m≥1$, we define (compare [ref] ) [equation] where [equation] Introduce the generating functions [equation]
2104.12336 · A Hall algebra realization of imaginary root vectors · 0.82
Cites as background 3 papers
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The $_i$ are directly related to $$Hall algebra.) The $q$-Onsager algebra $$ is the $$quantum group of split affine $A_1$ type, a special case of $$quantum groups in [ref] .
Hall algebras and quantum symmetric pairs of Kac-Moody type · The $q$-Onsager algebra · 0.80
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According to the $$program philosophy [ref] , $$quantum groups are viewed as a vast generalization of quantum groups, and various (algebraic, geometric, categorical) constructions on quantum groups should be generalizable to $$quantum groups.
A new approach to Kazhdan-Lusztig theory of type B via quantum symmetric pairs · Section · 0.90
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The $$Hall algebra of the Jordan quiver admits rich combinatorial properties as well, which will be studied in depth in a separate publication [ref] .
2104.12336 · Section · 0.97
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