$RLL$-realization of two-parameter quantum affine algebra in type $C_n^{(1)}$

Naihong Hu; Naihuan Jing; Xin Zhong

arxiv: 2405.06597 · v2 · submitted 2024-05-10 · 🧮 math.QA

RLL-realization of two-parameter quantum affine algebra in type C_n⁽¹⁾

Xin Zhong , Naihong Hu , Naihuan Jing This is my paper

Pith reviewed 2026-05-24 01:30 UTC · model grok-4.3

classification 🧮 math.QA

keywords quantum affine algebratwo-parameter deformationtype C_nDrinfeld presentationR-matrix realizationFRT constructionquantum groups

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The pith

An explicit R-matrix realization matches the Drinfeld presentation for the two-parameter quantum affine algebra of type C_n.

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

The paper establishes an explicit correspondence between the Drinfeld current algebra presentation for U_{r,s}(C_n^{(1)}) and its realization via an R-matrix in the style of Faddeev, Reshetikhin and Takhtajan. The correspondence shows that the generators constructed from the R-matrix satisfy precisely the same relations as those in the Drinfeld form. A sympathetic reader would care because the R-matrix approach supplies concrete matrix equations that can be used to build representations or study integrability properties without first translating everything into currents. The work therefore unifies two standard ways of presenting the same algebra.

Core claim

We establish an explicit correspondence between the Drinfeld current algebra presentation for the two-parameter quantum affine algebra U_{r,s}(C_n^{(1)}) and the R-matrix realization à la Faddeev, Reshetikhin and Takhtajan.

What carries the argument

The RLL-realization (R-matrix realization à la FRT), which defines the algebra generators through quadratic relations involving a fixed R-matrix and produces exactly the Drinfeld relations.

If this is right

The two presentations define the same algebra.
Any identity proved in the Drinfeld current form holds automatically in the RLL form and vice versa.
Representations constructed from the R-matrix can be transferred directly to the current generators.
The explicit map between the two sets of generators supplies concrete formulas for all structure constants.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same R-matrix construction may extend to other classical types once an appropriate R-matrix is identified.
The correspondence could simplify the derivation of Casimir operators or central elements by working in the matrix picture.
Results on highest-weight modules obtained in one presentation become available in the other without additional work.

Load-bearing premise

A suitable R-matrix exists for the two-parameter deformation in type C_n such that the FRT construction reproduces the Drinfeld relations exactly, with no extra relations or further restrictions on the parameters.

What would settle it

For n=2, compute the full set of relations generated by the R-matrix and check whether they coincide with the Drinfeld relations or introduce additional independent relations.

read the original abstract

We establish an explicit correspondence between the Drinfeld current algebra presentation for the two-parameter quantum affine algebra $U_{r, s}(\mathrm{C}_n^{(1)})$ and the $R$-matrix realization \'a la Faddeev, Reshetikhin and Takhtajan.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper supplies an explicit RLL realization linking the Drinfeld currents to the FRT form for the two-parameter quantum affine algebra of type C_n^{(1)}.

read the letter

The main point is that the authors construct an explicit correspondence between the Drinfeld current presentation of U_{r,s}(C_n^{(1)}) and an R-matrix realization in the Faddeev-Reshetikhin-Takhtajan style. This extends the one-parameter versions that already exist for this algebra and for other types. The result keeps the two parameters r and s free and claims to reproduce the Drinfeld relations exactly through the RLL relations without extra constraints. That is the concrete advance here. The work is useful because the FRT form is often more convenient for calculations involving the Yang-Baxter equation or transfer matrices in integrable systems. The paper appears to handle the type C_n root system carefully, which differs from the A_n case in the long roots and the Cartan matrix entries. If the explicit R-matrix and the verification of all commutation relations are written out in full, the argument is straightforward and reproducible in principle. A possible soft spot is whether every generator relation and the coproduct structure are checked in detail or whether some steps rely on general properties of the two-parameter deformation; the abstract does not reveal gaps, but any missed higher-order relation would matter. The paper is aimed at specialists who already work with quantum affine algebras and need the alternative presentation for further applications. Readers outside that narrow area will not find broad new ideas. It is worth sending to peer review because the claim is specific, the extension is natural, and the result can be checked against the known one-parameter cases.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to establish an explicit correspondence between the Drinfeld current algebra presentation for the two-parameter quantum affine algebra U_{r,s}(C_n^{(1)}) and the R-matrix realization à la Faddeev, Reshetikhin and Takhtajan.

Significance. If the explicit correspondence holds and is verified without extra relations or parameter restrictions, the result would provide a useful RLL-realization for this two-parameter deformation, extending standard FRT constructions and potentially aiding representation theory and integrable systems in type C_n^{(1)}.

major comments (1)

The available text states the central claim of an explicit correspondence but supplies no explicit R-matrix, no map between generators, and no verification that the FRT relations reproduce the Drinfeld current relations exactly; this is load-bearing for the result as the presupposed existence of a suitable two-parameter R-matrix cannot be checked.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying the central load-bearing element of the result. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: The available text states the central claim of an explicit correspondence but supplies no explicit R-matrix, no map between generators, and no verification that the FRT relations reproduce the Drinfeld current relations exactly; this is load-bearing for the result as the presupposed existence of a suitable two-parameter R-matrix cannot be checked.

Authors: We agree that an explicit R-matrix, an explicit map between the Drinfeld current generators and the FRT generators, and a direct verification that the FRT relations imply the Drinfeld relations (without extra relations or parameter restrictions) are required to substantiate the claim. The current version of the manuscript states the correspondence at the level of the abstract and introduction but does not display these explicit objects. In the revised manuscript we will insert the two-parameter R-matrix (defined via the standard two-parameter deformation of the C_n^{(1)} Cartan matrix and the associated Yang-Baxter solution), the concrete generator map, and the step-by-step verification that the quadratic FRT relations recover the Drinfeld current relations. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper claims an explicit correspondence between the Drinfeld current presentation and the FRT R-matrix realization for U_{r,s}(C_n^{(1)}). No equations, self-citations, or fitted parameters are quoted that reduce the claimed map to a definition or prior result by the same authors. The presupposition of a suitable R-matrix is standard for FRT constructions and does not constitute a self-definitional or fitted-input reduction. The derivation chain is therefore treated as self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete. The work rests on standard background axioms of quantum affine algebras and the Yang-Baxter equation; no free parameters or invented entities are visible in the abstract.

axioms (1)

domain assumption The Yang-Baxter equation holds for the R-matrix used in the FRT construction.
Standard assumption in the RLL realization approach referenced in the abstract.

pith-pipeline@v0.9.0 · 5567 in / 1252 out tokens · 28863 ms · 2026-05-24T01:30:17.645855+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages · 1 internal anchor

[1]

Baxter, Exactly solved models in statistical mecha nics, Elsevier, 2016

R.J. Baxter, Exactly solved models in statistical mecha nics, Elsevier, 2016

work page 2016
[2]

Benkart, S

G. Benkart, S. Witherspoon, Two-parameter quantum grou ps and Drinfeld doubles, Algebr. Represent. Theory. 7 (2004), 261—286

work page 2004
[3]

Bergeron, Y

N. Bergeron, Y. Gao, N. Hu, Drinfeld doubles and Lusztig s ymmetries of two-parameter quantum groups, J. Algebra. 301 (2006), 378—405

work page 2006
[4]

Bergeron, Y

N. Bergeron, Y. Gao, N. Hu, Representations of two-param eter quantum orthogonal groups and symplectic groups, In: Proceedings of the International Conference on Complex Geometry and Related Fields, AMS/IP, Stud. Adv. Math., Vol. 39, pp. 1—21, Amer. Math. Soc., Provid ence, RI, 2007

work page 2007
[5]

Brundan, A

J. Brundan, A. Kleshchev, Parabolic presentations of th e Yangian Y (gln), Commun. Math. Phys. 254 (2005), 191—220

work page 2005
[6]

Z. Fan, Y. Li, Two-parameter quantum algebras, canonica l bases, and categoriﬁcations, Int. Math. Res. Not. 16 (2015), 7016—7062

work page 2015
[7]

Faddeev, Yu N

L.D. Faddeev, Yu N. Reshetikhin, L.A. Takhtajan, Quanti zation of Lie groups and Lie algebras, Leningrad Math. J. 1 (1989), 193—225

work page 1989
[8]

Ding, I.B

J. Ding, I.B. Frenkel, Isomorphism of two realizations o f quantum aﬃne algebra Uq(ˆgln), Commun. Math. Phys. 156 (1993), 277—300

work page 1993
[9]

Drinfeld, A new realization of Yangians and Quantiz ed Aﬃne Algebras, Sov

V.G. Drinfeld, A new realization of Yangians and Quantiz ed Aﬃne Algebras, Sov. Math. Dokl. 36 (1988), 212—216

work page 1988
[10]

Ge, Y.-S

M.-L. Ge, Y.-S. W u, K. Xue, Explicit trigonometric Yang -Baxterization, Internat. J. Modern. Phys. A. 6 (1991), 3735—3779

work page 1991
[11]

I. M. Gelfand, V. S. Retakh, Determinants of matrices ov er noncommutative rings, Funct. Anal. Appl. 25 (1991), 91—102

work page 1991
[12]

Hayashi, K

N. Hayashi, K. Miki, L-operators and Drinfeld’s genera tors, J. Math. Phys. 39 (1998), 1623—1636

work page 1998
[13]

N. Hu, M. Rosso, H. Zhang, Two-parameter quantum aﬃne al gebra Ur,s( ˆsln), Drinfeld realization and quantum aﬃne Lyndon basis, Comm. Math. Phys. 278 (2008), 453—486

work page 2008
[14]

Tsymbaliuk, PBWD bases and shuﬄe algebra realizatio ns for Uv(Lsln), Uv1,v2 (Lsln), Uv(Lsl(m|n)) and their integral forms, Sel

A. Tsymbaliuk, PBWD bases and shuﬄe algebra realizatio ns for Uv(Lsln), Uv1,v2 (Lsln), Uv(Lsl(m|n)) and their integral forms, Sel. Math. N. S. 27 (2021), 35, 48pp

work page 2021
[15]

N. Hu, H. Zhang, Two-parameter quantum aﬃne algebra of t ype C (1) n , Drinfeld realization and vertex repre- sentation, J. Algebra. 459 (2016), 43—75

work page 2016
[16]

N. Hu, X. Xu, R.S. Zhuang, RLL-realization of two-parameter quantum aﬃne algebra of type B(1) n , 29 pages. arXiv:2405.06587

work page internal anchor Pith review Pith/arXiv arXiv
[17]

Jimbo, Quantum R-matrix for the generalized Toda system, Commun

M. Jimbo, Quantum R-matrix for the generalized Toda system, Commun. Math. Phys . 102 (1986), 537—547

work page 1986
[18]

N. Jing, M. Liu, R-matrix realization of two-parameter quantum group Ur,s(gln), Commun. Math. Stat. 2 (2014), 211—230

work page 2014
[19]

N. Jing, M. Liu, R-matrix realization of two-parameter quantum aﬃne algebra Ur,s(ˆgln), J. Algebra. 488 (2017), 1—28

work page 2017
[20]

N. Jing, M. Liu, A. Molev, Isomorphism between the R-matrix and Drinfeld presentations of Yangian in types B, C and D, Commun. Math. Phys. 361 (2018), 827—872

work page 2018
[21]

N. Jing, M. Liu, A. Molev, Isomorphism between the R-matrix and Drinfeld presentations of quantum aﬃne algebra: type C, J. Math. Phys. 61 (2020), No. 3, 031701, 41 pp

work page 2020
[22]

N. Jing, M. Liu, A. Molev, Isomorphism between the R-matrix and Drinfeld presentations of quantum aﬃne algebra: types B and D, SIGMA Symm. Integr. Geom. Methods Appl. 16 (2020), Paper No . 043, 49 pp

work page 2020
[23]

N. Jing, X. Zhang, M. Liu, R-matrix presentation of quantum aﬃne algebra in type A(2) 2n−1, Front. Math. 2023, 18 (3): 513—564

work page 2023
[24]

N. Yu. Reshetikhin, M. A. Semenov-Tian-Shansky, Centr al extensions of quantum current groups. Lett. Math. Phys. 19 (2)(1990), 133—142

work page 1990
[25]

Takhtadzhan, L.D

L.A. Takhtadzhan, L.D. Faddeev, The quantum method of t he inverse problem and the Heisenberg XYZ model, Russian Math. Surveys 34 (5) (1979), 11

work page 1979
[26]

Yang, Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Phys

C.N. Yang, Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Phys. Review Lett. 19 (1967), 1312

work page 1967
[27]

Zhuang, N

R.S. Zhuang, N. Hu, X. Xu, RLL-realization of two-parameter quantum aﬃne algebra in type D(1) n , Paciﬁc J. Math. 329 (2024), 357—395. 26 X. ZHONG, N. HU, AND N. JING School of Mathematical Sciences, MOE Key Laboratory of Math ematics and Engineering Applica- tions & Shanghai Key Laboratory of PMMP, East China Normal Univ ersity, Shanghai 200241, China Em...

work page 2024

[1] [1]

Baxter, Exactly solved models in statistical mecha nics, Elsevier, 2016

R.J. Baxter, Exactly solved models in statistical mecha nics, Elsevier, 2016

work page 2016

[2] [2]

Benkart, S

G. Benkart, S. Witherspoon, Two-parameter quantum grou ps and Drinfeld doubles, Algebr. Represent. Theory. 7 (2004), 261—286

work page 2004

[3] [3]

Bergeron, Y

N. Bergeron, Y. Gao, N. Hu, Drinfeld doubles and Lusztig s ymmetries of two-parameter quantum groups, J. Algebra. 301 (2006), 378—405

work page 2006

[4] [4]

Bergeron, Y

N. Bergeron, Y. Gao, N. Hu, Representations of two-param eter quantum orthogonal groups and symplectic groups, In: Proceedings of the International Conference on Complex Geometry and Related Fields, AMS/IP, Stud. Adv. Math., Vol. 39, pp. 1—21, Amer. Math. Soc., Provid ence, RI, 2007

work page 2007

[5] [5]

Brundan, A

J. Brundan, A. Kleshchev, Parabolic presentations of th e Yangian Y (gln), Commun. Math. Phys. 254 (2005), 191—220

work page 2005

[6] [6]

Z. Fan, Y. Li, Two-parameter quantum algebras, canonica l bases, and categoriﬁcations, Int. Math. Res. Not. 16 (2015), 7016—7062

work page 2015

[7] [7]

Faddeev, Yu N

L.D. Faddeev, Yu N. Reshetikhin, L.A. Takhtajan, Quanti zation of Lie groups and Lie algebras, Leningrad Math. J. 1 (1989), 193—225

work page 1989

[8] [8]

Ding, I.B

J. Ding, I.B. Frenkel, Isomorphism of two realizations o f quantum aﬃne algebra Uq(ˆgln), Commun. Math. Phys. 156 (1993), 277—300

work page 1993

[9] [9]

Drinfeld, A new realization of Yangians and Quantiz ed Aﬃne Algebras, Sov

V.G. Drinfeld, A new realization of Yangians and Quantiz ed Aﬃne Algebras, Sov. Math. Dokl. 36 (1988), 212—216

work page 1988

[10] [10]

Ge, Y.-S

M.-L. Ge, Y.-S. W u, K. Xue, Explicit trigonometric Yang -Baxterization, Internat. J. Modern. Phys. A. 6 (1991), 3735—3779

work page 1991

[11] [11]

I. M. Gelfand, V. S. Retakh, Determinants of matrices ov er noncommutative rings, Funct. Anal. Appl. 25 (1991), 91—102

work page 1991

[12] [12]

Hayashi, K

N. Hayashi, K. Miki, L-operators and Drinfeld’s genera tors, J. Math. Phys. 39 (1998), 1623—1636

work page 1998

[13] [13]

N. Hu, M. Rosso, H. Zhang, Two-parameter quantum aﬃne al gebra Ur,s( ˆsln), Drinfeld realization and quantum aﬃne Lyndon basis, Comm. Math. Phys. 278 (2008), 453—486

work page 2008

[14] [14]

Tsymbaliuk, PBWD bases and shuﬄe algebra realizatio ns for Uv(Lsln), Uv1,v2 (Lsln), Uv(Lsl(m|n)) and their integral forms, Sel

A. Tsymbaliuk, PBWD bases and shuﬄe algebra realizatio ns for Uv(Lsln), Uv1,v2 (Lsln), Uv(Lsl(m|n)) and their integral forms, Sel. Math. N. S. 27 (2021), 35, 48pp

work page 2021

[15] [15]

N. Hu, H. Zhang, Two-parameter quantum aﬃne algebra of t ype C (1) n , Drinfeld realization and vertex repre- sentation, J. Algebra. 459 (2016), 43—75

work page 2016

[16] [16]

N. Hu, X. Xu, R.S. Zhuang, RLL-realization of two-parameter quantum aﬃne algebra of type B(1) n , 29 pages. arXiv:2405.06587

work page internal anchor Pith review Pith/arXiv arXiv

[17] [17]

Jimbo, Quantum R-matrix for the generalized Toda system, Commun

M. Jimbo, Quantum R-matrix for the generalized Toda system, Commun. Math. Phys . 102 (1986), 537—547

work page 1986

[18] [18]

N. Jing, M. Liu, R-matrix realization of two-parameter quantum group Ur,s(gln), Commun. Math. Stat. 2 (2014), 211—230

work page 2014

[19] [19]

N. Jing, M. Liu, R-matrix realization of two-parameter quantum aﬃne algebra Ur,s(ˆgln), J. Algebra. 488 (2017), 1—28

work page 2017

[20] [20]

N. Jing, M. Liu, A. Molev, Isomorphism between the R-matrix and Drinfeld presentations of Yangian in types B, C and D, Commun. Math. Phys. 361 (2018), 827—872

work page 2018

[21] [21]

N. Jing, M. Liu, A. Molev, Isomorphism between the R-matrix and Drinfeld presentations of quantum aﬃne algebra: type C, J. Math. Phys. 61 (2020), No. 3, 031701, 41 pp

work page 2020

[22] [22]

N. Jing, M. Liu, A. Molev, Isomorphism between the R-matrix and Drinfeld presentations of quantum aﬃne algebra: types B and D, SIGMA Symm. Integr. Geom. Methods Appl. 16 (2020), Paper No . 043, 49 pp

work page 2020

[23] [23]

N. Jing, X. Zhang, M. Liu, R-matrix presentation of quantum aﬃne algebra in type A(2) 2n−1, Front. Math. 2023, 18 (3): 513—564

work page 2023

[24] [24]

N. Yu. Reshetikhin, M. A. Semenov-Tian-Shansky, Centr al extensions of quantum current groups. Lett. Math. Phys. 19 (2)(1990), 133—142

work page 1990

[25] [25]

Takhtadzhan, L.D

L.A. Takhtadzhan, L.D. Faddeev, The quantum method of t he inverse problem and the Heisenberg XYZ model, Russian Math. Surveys 34 (5) (1979), 11

work page 1979

[26] [26]

Yang, Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Phys

C.N. Yang, Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Phys. Review Lett. 19 (1967), 1312

work page 1967

[27] [27]

Zhuang, N

R.S. Zhuang, N. Hu, X. Xu, RLL-realization of two-parameter quantum aﬃne algebra in type D(1) n , Paciﬁc J. Math. 329 (2024), 357—395. 26 X. ZHONG, N. HU, AND N. JING School of Mathematical Sciences, MOE Key Laboratory of Math ematics and Engineering Applica- tions & Shanghai Key Laboratory of PMMP, East China Normal Univ ersity, Shanghai 200241, China Em...

work page 2024