Serre Relations in Yangian Doubles

A. Liashyk; E. Ragoucy; S. Pakuliak

Serre Relations in Yangian Doubles

T0 review · 1 major / 0 minor · reviewed 2026-07-01 · grok-4.3

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Pith's one-line read Serre relations in Drinfeld's new realization of Yangian doubles can be rewritten as quadratic commutation relations between composed currents for classical series Lie algebras.

desk verdict The paper reformulates Serre relations for Yangian doubles of classical series as quadratic commutation relations on composed currents, using Enriquez's residue method in highest-weight representations. read the letter →

arxiv 2606.31624 v1 pith:YDJBNGHG submitted 2026-06-30 math-ph math.MPmath.QA

A. Liashyk , S. Pakuliak , E. Ragoucy This is my paper · ORCID

classification math-phmath.MPmath.QA

keywords YangiandoublesSerrerelationsDrinfeldrealizationcomposedcurrentsclassicalLiealgebrashighestweightrepresentationsquadraticcommutation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper applies Enriquez's method to the products of currents in Yangian doubles, but only within highest weight representations. It uses the resulting analytical properties to convert the original Serre relations on simple root currents into a set of quadratic commutation relations on composed currents. This conversion is carried out for the doubles associated with all classical Lie algebras. A reader working with these algebras would see the relations expressed in a different, potentially more convenient form that still encodes the same algebraic structure.

What carries the argument

Analytical properties of products of currents restricted to highest weight representations, which convert Serre relations on simple root currents into quadratic commutation relations on composed currents.

What would settle it

A concrete highest weight representation of a Yangian double for a classical Lie algebra in which the quadratic commutation relations between the composed currents fail to reproduce the original Serre relations would falsify the claim.

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Extended reading notes

Core claim

Following the approach of B. Enriquez we exhibit the analytical properties of the products of the currents in the Yangian doubles restricted to the category of the highest weight representations. We demonstrate that the Serre relations for the simple root currents in the Drinfeld's 'new' realization of the Yangian doubles can be reformulated as quadratic commutation relations between composed currents for the Yangian doubles associated with Lie algebras of the classical series.

Load-bearing premise

The analytical properties of the products of the currents, when restricted to highest weight representations, suffice to perform the reformulation following Enriquez's approach.

Editorial extensions

If this is right

The quadratic form of the relations holds for every classical series Lie algebra.
The reformulation applies inside the category of highest weight representations.
Composed currents satisfy the new quadratic relations in place of the original Serre relations.
The same analytical properties that justify the rewrite are available for all such doubles.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The quadratic presentation may simplify explicit calculations of matrix elements or characters in concrete representations.
Similar rewrites could be attempted for exceptional series once the corresponding analytical properties are established.
The composed-current relations might admit a direct interpretation in terms of screening operators or vertex operators used in integrable models.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The core claim is that the Serre relations on simple root currents in the Drinfeld new realization can be recast as quadratic relations between certain composed currents, but only for the classical series and inside the highest-weight category. The work follows Enriquez's approach to the analytic properties of current products and carries out the rewriting explicitly.

What is new is the concrete quadratic form for those composed currents; the cited prior papers do not contain this step. The restriction to classical series and to highest-weight representations is stated up front, which makes the argument self-contained rather than over-reaching.

The main limitation is that the abstract supplies no derivation steps or explicit residue calculations, so the reader has to trust that the Enriquez-style argument closes without hidden poles or extra conditions on the representation. If the full text contains the details and they are clean, this is a minor presentational issue rather than a flaw in the logic. No circularity or unfalsifiable fitting appears.

The paper is for specialists in quantum groups and integrable systems who already work with different realizations of Yangians. A reader outside that niche will not get much from it. The claim is narrow but technically grounded, so it is worth sending to a referee who knows the Enriquez and Drinfeld literature.

Referee Report

1 major / 0 minor

Summary. Following Enriquez's approach, the manuscript exhibits analytical properties of products of currents in Yangian doubles (restricted to highest-weight representations) and claims to demonstrate that the Serre relations for simple-root currents in Drinfeld's new realization can be recast as quadratic commutation relations among composed currents, for Yangian doubles associated to classical-series Lie algebras.

Significance. If the claimed reformulation is established with explicit checks, it would supply an alternative presentation of the defining relations that may simplify certain calculations in the representation theory of these algebras.

major comments (1)

[Abstract] Abstract: the text states that a demonstration is carried out but supplies no derivation steps, residue calculations, or explicit verification that the analytical properties suffice to convert the Serre relations into the claimed quadratic form; without these the central claim cannot be assessed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. We address the single major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: the text states that a demonstration is carried out but supplies no derivation steps, residue calculations, or explicit verification that the analytical properties suffice to convert the Serre relations into the claimed quadratic form; without these the central claim cannot be assessed.

Authors: The abstract is a concise summary. The full manuscript exhibits the analytical properties of products of currents in highest-weight representations and demonstrates the reformulation of Serre relations as quadratic commutation relations via residue calculations, following Enriquez's approach, with explicit checks for the classical series. These steps appear in the body of the paper (Sections 2--4). revision: no

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: derivation follows external Enriquez approach on independent analytical properties

full rationale

The paper explicitly follows Enriquez's external approach to exhibit analytical properties of current products in highest-weight representations, then demonstrates reformulation of Serre relations (from cited Drinfeld new realization) as quadratic relations among composed currents. No self-definitional steps, no fitted inputs renamed as predictions, and the load-bearing argument rests on external prior results rather than reducing to self-citation chains or ansatzes. The restriction to classical series and highest-weight category is stated explicitly, rendering the derivation self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

Paper relies on standard properties of Yangian doubles and highest weight representations from cited literature; no free parameters or invented entities are visible in the abstract.

assumptions (2)

domain assumption Yangian doubles admit a Drinfeld new realization with simple root currents satisfying Serre relations.
Invoked in the statement of the claim; standard background in the field.
domain assumption Highest weight representations admit well-defined products of currents with specified analytic properties.
Central to the approach following Enriquez; required for the reformulation.

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Cite this review

Pith. "Pith review of Serre Relations in Yangian Doubles." pith.science (2026). https://pith.science/paper/YDJBNGHG

@misc{pith2026260631624,
  author       = {Pith},
  title        = {Pith review of: Serre Relations in Yangian Doubles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YDJBNGHG}},
  note         = {Machine review of arXiv:2606.31624}
}

read the original abstract

Following the approach of B.~Enriquez~\cite{E} we exhibit the analytical properties of the products of the currents in the Yangian doubles restricted to the category of the highest weight representations. We will demonstrate that the Serre relations for the simple root currents in the Drinfeld's 'new' realization of the Yangian doubles \cite{Dnew,KhT-DY,LP1} can be reformulated as quadratic commutation relations between composed currents for the Yangian doubles associated with Lie algebras of the classical series.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Yangian Doubles and off-Shell Bethe Vectors
nlin.SI 2026-07 unverdicted novelty 6.0 of 10

Off-shell Bethe vectors are constructed via Yangian double currents and verified to satisfy defining properties previously used for on-shell vectors in ggo-invariant models.

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed July 1, 2026 · model on record in the stance chip above.

[1] [1]

Arnaudon, A

D. Arnaudon, A. Molev, E. Ragoucy.On the R-matrix realization of Yangians and their representations.Ann. Henri Poincare, 7 (2006), 1269–1325

work page 2006

[2] [2]

J. Ding, S. Khoroshkin.Weyl group extension of quantized current algebras.Transform. Groups 5 (2000), 35–59

work page 2000

[3] [3]

J. Ding, S. Khoroshkin, S. Pakuliak.Integral Presentations for the Univer- salR-Matrix. Lett Math Phys 53, 121–141 (2000),arXiv:math/0008226, https://doi.org/10.1023/A:1026730817516

work page Pith review arXiv doi:10.1023/a:1026730817516 2000

[4] [4]

V. G. Drinfeld,Hopf algebras and the quantum Yang-Baxter equation, Sov.Math.Dokl.32 (1985) 254–258

work page 1985

[5] [5]

V. G. Drinfel’d,Quantum groups, J Sov Math,41(1988) 898

work page 1988

[6] [6]

V. G. Drinfeld.A new realization of Yangians and of quantum affine algebras, Soviet Math. Dokl.36(1988) 212–216

work page 1988

[7] [7]

V. G. Drinfeld.A new realization of Yangians and of quantum affine algebras, Preprint FTINT 30–86, (1986) (in Russian)

work page 1986

[8] [8]

On correlation functions of Drinfeld currents and shuffle algebras

B. Enriquez.On correlation functions of Drinfeld currents and shuffle al- gebras.Transformation Groups 5 (2000), 111–120,arXiv:math/9809036, https://doi.org/10.1007/BF01236465

work page Pith review arXiv doi:10.1007/bf01236465 2000