Quantum Algebra · arXiv:2103.11229 v1 · submitted 20 Mar 2021
The compact presentation for the alternating central extension of the q-Onsager algebra
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“In this paper we obtain a presentation of 𝒪q that involves a subset of the original set of generators and a very manageable set of relations.”
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Abstract
The q-Onsager algebra Oq is defined by two generators and two relations, called the q-Dolan/Grady relations. We investigate the alternating central extension 𝒪q of Oq. The algebra 𝒪q was introduced by Baseilhac and Koizumi, who called it the current algebra of Oq. Recently Baseilhac and Shigechi gave a presentation of 𝒪q by generators and relations. The presentation is attractive, but the multitude of generators and relations makes the presentation unwieldy. In this paper we obtain a presentation of 𝒪q that involves a subset of the original set of generators and a very manageable set of relations. We call this presentation the compact presentation of 𝒪q. This presentation resembles the compact presentation of the alternating central extension for the positive part of $U_q(\widehat{\mathfrak{sl}}_2)$.
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abstract compiled · 2026-09-23 · no language-model review
How the main claim is stated: asserted 1.00
What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.10 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.96 · mentions code or data availability 0.02
The abstract, sentence by sentence
This table labels the abstract only (arXiv listing abstract v1). The number is the probability the classifier gave the label.
| Sentence | Role | The authors' own work | States the main conclusion | Strength |
|---|---|---|---|---|
| The $q$-Onsager algebra $O_q$ is defined by two generators and two relations, called the $q$-Dolan/Grady relations. | background 1.00 | 0.06 | 0.02 | asserted 1.00 |
| We investigate the alternating central extension $\mathcal O_q$ of $O_q$. | contribution 0.81 | 0.66 | 0.04 | asserted 0.94 |
| The algebra $\mathcal O_q$ was introduced by Baseilhac and Koizumi, who called it the current algebra of $O_q$. | background 1.00 | 0.04 | 0.02 | asserted 1.00 |
| Recently Baseilhac and Shigechi gave a presentation of $\mathcal O_q$ by generators and relations. | background 1.00 | 0.04 | 0.03 | asserted 1.00 |
| The presentation is attractive, but the multitude of generators and relations makes the presentation unwieldy. | background 0.55 | 0.17 | 0.03 | asserted 1.00 |
| In this paper we obtain a presentation of $\mathcal O_q$ that involves a subset of the original set of generators and a very manageable set of relations. | contribution 0.89 | 0.96 | 0.87 | asserted 1.00 |
| We call this presentation the compact presentation of $\mathcal O_q$. | contribution 0.71 | 0.89 | 0.21 | asserted 1.00 |
| This presentation resembles the compact presentation of the alternating central extension for the positive part of $U_q(\widehat{\mathfrak{sl}}_2)$. | interpretation 0.82 | 0.32 | 0.08 | asserted 0.99 |
What kind of paper the classifier reports this is: theoretical result 0.98 · evidence from proof or derivation 0.99
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Cited by 1 Pith paper
Reviewed papers on Pith that cite this one. The full citation record
How the literature uses it
5 arXiv papers cite this in 12 sentences
Papers citing it · 5 sentences without a confident label
Compares with it 1 paper
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The analogous presentations of $𝒪_q$ can be found in [ref] and [ref] .
The alternating central extension of the Onsager Lie algebra · How $O$ and $𝒪$ are related to $O_q$ and $𝒪_q$ · 0.84
Cites it as background 5 papers
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As discussed in the series of works [ref] , $O_q$ is obtained by applying a central reduction of the algebra $A_q$.
Universal TT- and TQ-relations via centrally extended q-Onsager algebra · The case of the q-Onsager algebra · 0.89
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This paper is part of a sequence [ref] , [ref] , [ref] concerning the $q$-Onsager algebra and its alternating central extension.
The $q$-Onsager algebra and its alternating central extension · Introduction · 0.87
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The alternating central extension of $O_q$ is discussed in [ref] .
The $S_3$-symmetric $q$-Onsager algebra and its Lusztig automorphisms · Introduction · 1.00
What it cites · 58 sentences without a confident label
Uses the method or tool of 1 paper
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Following [ref] we use [ref], [ref] to recursively obtain $𝒲_-k$, $𝒲_k+1$ for $k≥1$: [equation] [equation]
An action of the free product $\mathbb Z_2 \star \mathbb Z_2 \star \mathbb Z_2$ on the $q$-Onsager algebra and its current algebra · The essential generators for $𝒪_q$ · 1.00
Compares with 1 paper
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In [ref] Baseilhac gives a similar injective algebra homomorphism $𝒰^+_q →U_q(𝔤𝔩_2)$.
The alternating presentation of $U_q(\widehat{gl_2})$ from Freidel-Maillet algebras · Introduction · 0.91
Cites as support 1 paper
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We have [equation] and [equation] Moreover for $n≥1$, [equation] By [ref] the elements ${B_nδ}_n=1^∞$ mutually commute.
Braid group action and root vectors for the $q$-Onsager algebra · The elements ${𝒲_-k}_k ∈ℕ$, ${𝒲_k+1}_k ∈ℕ$ revisited · 0.97
Cites as background 17 papers
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In [ref] Baseilhac and Shigechi give a presentation of $𝒪_q$ by generators and relations.
A new current algebra and the reflection equation · Introduction · 0.85
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(See [ref] , [ref] .) Define the algebra $𝒪_q$ by generators [equation] and the following relations.
A new (in)finite dimensional algebra for quantum integrable models · The algebra $𝒪_q$ · 0.96
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We recall from [ref] a filtration of $O_q$.
The $q$-Onsager algebra and the positive part of $U_q({\widehat{\mathfrak{sl}}}_2)$ · A filtration of $𝒪_q$, cont. · 0.84
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