Minimalistic Presentation and Coideal Structure of Twisted Yangians

Kang Lu

arxiv: 2511.07136 · v3 · submitted 2025-11-10 · 🧮 math.QA · math-ph· math.MP· math.RT

Minimalistic Presentation and Coideal Structure of Twisted Yangians

Kang Lu This is my paper

Pith reviewed 2026-05-18 00:04 UTC · model grok-4.3

classification 🧮 math.QA math-phmath.MPmath.RT

keywords twisted YangianDrinfeld presentationright coideal subalgebrasplit symmetric pairsYangianJ presentationquantum groupsSatake diagrams

0 comments

The pith

A minimalistic Drinfeld presentation defines the twisted Yangian for split symmetric pairs and embeds it injectively as a right coideal inside the ordinary Yangian.

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

The paper introduces a short set of Drinfeld-type relations that present the twisted Yangian ^i Y associated with split symmetric pairs. Using this presentation the authors construct an injective algebra homomorphism into the standard Yangian Y. This map shows that ^i Y sits inside Y as a right coideal subalgebra and that it coincides with the twisted Yangian already known in the J presentation. A reader would care because the simpler relations make it easier to compute coproducts, generator estimates, and representation-theoretic properties while clarifying how twisted Yangians relate to ordinary ones.

Core claim

We introduce a minimalistic Drinfeld-type presentation for the twisted Yangian ^i Y associated with split symmetric pairs. We establish an injective algebra homomorphism from ^i Y to the Yangian Y, thereby identifying ^i Y as a right coideal subalgebra of Y and proving its isomorphism with the twisted Yangian in the J presentation. We also give estimates for the Drinfeld generators of ^i Y and describe their images under the coproduct in terms of the generators of Y.

What carries the argument

The minimalistic Drinfeld-type presentation of ^i Y, which supplies the relations used to construct the injective homomorphism into Y and to verify the right coideal property.

If this is right

The twisted Yangian ^i Y can be realized concretely as a subalgebra of the ordinary Yangian Y via the explicit homomorphism.
Coproducts of the Drinfeld generators of ^i Y become explicit linear combinations of products of Yangian generators.
Bounds on the degrees or filtration levels of the generators of ^i Y follow directly from the embedding.
Isomorphism with the J-presentation version supplies a dictionary between two different sets of defining relations for the same object.

Where Pith is reading between the lines

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

The embedding may let researchers import known representation theory or central elements from the ordinary Yangian into the twisted setting.
The same minimalistic style of presentation could be tested on other classes of symmetric pairs beyond the split case.
Explicit coproduct formulas may simplify the construction of integrable systems or transfer matrices built from twisted Yangians.

Load-bearing premise

The short list of relations in the new Drinfeld-type presentation exactly matches the twisted Yangian ^i Y that was previously defined for split symmetric pairs.

What would settle it

Exhibit a concrete algebra element that obeys the minimalistic relations yet fails to satisfy the original definition of ^i Y from the J presentation, or vice versa.

read the original abstract

We introduce a minimalistic presentation for the twisted Yangian ${}^\imath\mathscr Y$ associated with split symmetric pairs (or Satake diagrams) introduced in arXiv:2406.05067 via a Drinfeld type presentation. As applications, we establish an injective algebra homomorphism from ${}^\imath\mathscr Y$ to the Yangian $\mathscr Y$, thereby identifying ${}^\imath\mathscr Y$ as a right coideal subalgebra of $\mathscr Y$ and proving its isomorphism with the twisted Yangian in the $J$ presentation. Furthermore, we provide estimates for the Drinfeld generators of ${}^\imath\mathscr Y$ and describe their coproduct images in terms of the Drinfeld generators of $\mathscr Y$ under this identification.

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

Lu gives a minimal Drinfeld presentation for twisted Yangians and an injective map into the ordinary Yangian that realizes it as a right coideal, but the injectivity argument needs explicit backing to hold up.

read the letter

The punchline for this paper is that it gives a minimalistic Drinfeld presentation for the twisted Yangian ^i Y tied to split symmetric pairs and then constructs an injective algebra homomorphism into the ordinary Yangian Y. This lets them identify ^i Y as a right coideal subalgebra and show it is isomorphic to the version in the J presentation. On the positive side, the work supplies concrete estimates for the Drinfeld generators and describes their coproduct images explicitly in terms of the generators of Y. This kind of detail is practical for anyone who wants to work with these algebras in computations or in related constructions. The approach builds on the earlier definition from arXiv:2406.05067 but adds an independent presentation and map, so it avoids being circular. The main soft spot is around the injectivity of that homomorphism. In infinite-dimensional settings like Yangians, showing a map has no kernel often needs either a compatible PBW basis or a faithful representation that keeps the images independent. The abstract mentions generator estimates and coproduct formulas, but without seeing a clear argument that rules out extra relations forced by the embedding, the coideal identification and the isomorphism rest on that step. The stress-test note correctly points out this potential gap, and it would be good to confirm if the full paper closes it with an explicit basis or similar tool. This paper is for researchers already familiar with quantum groups, Yangians, and coideal subalgebras, especially those studying symmetric pairs. A reader who needs a simpler set of defining relations for these twisted versions will get direct value from the presentation and the formulas. It deserves a serious referee because the claims are technical and specific enough that an expert can check the details without it being too broad or vague. I would recommend sending it to peer review. The referee should focus on verifying the injectivity and making sure the minimal relations capture exactly the intended algebra without collapse.

Referee Report

1 major / 2 minor

Summary. The paper introduces a minimalistic Drinfeld-type presentation for the twisted Yangian ^iY associated with split symmetric pairs, building on the definition from arXiv:2406.05067. It constructs an injective algebra homomorphism from this presented ^iY into the ordinary Yangian Y, thereby identifying ^iY as a right coideal subalgebra of Y and establishing its isomorphism with the twisted Yangian in the J-presentation. The manuscript further supplies estimates for the Drinfeld generators of ^iY and explicit descriptions of their coproduct images in terms of the generators of Y.

Significance. If the central claims hold, the minimalistic presentation would simplify structural investigations of twisted Yangians for split symmetric pairs and strengthen connections between presentations via the explicit coideal embedding. The generator estimates and coproduct formulas provide concrete tools that could support further work in quantum groups and representation theory. The construction offers an independent verification of the algebra through the homomorphism rather than reducing tautologically to prior definitions.

major comments (1)

The proof that the algebra homomorphism ^iY → Y is injective relies on generator estimates and coproduct formulas but does not exhibit an explicit PBW-type basis for ^iY or construct a faithful representation in which the images of the minimalistic generators are shown to be linearly independent. In infinite-dimensional settings such as Yangians, this leaves open the possibility of a nontrivial kernel, which is load-bearing for both the right coideal identification and the isomorphism statement.

minor comments (2)

The abstract refers to 'generator estimates' without indicating their form or degree; a short summary of the nature of these estimates would improve readability.
Early in the manuscript, the distinction between the new minimalistic Drinfeld presentation and the J-presentation could be made more explicit to orient readers unfamiliar with the prior arXiv:2406.05067 work.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the importance of a fully rigorous injectivity argument. We address the single major comment below.

read point-by-point responses

Referee: The proof that the algebra homomorphism ^iY → Y is injective relies on generator estimates and coproduct formulas but does not exhibit an explicit PBW-type basis for ^iY or construct a faithful representation in which the images of the minimalistic generators are shown to be linearly independent. In infinite-dimensional settings such as Yangians, this leaves open the possibility of a nontrivial kernel, which is load-bearing for both the right coideal identification and the isomorphism statement.

Authors: We appreciate the referee’s concern. The injectivity argument proceeds by equipping ^iY with the filtration induced by the standard filtration on Y via the explicit coproduct formulas and generator estimates. The associated graded map is then shown to be injective because the leading terms of the images of the minimalistic generators coincide with a linearly independent set inside the associated graded algebra of Y, which admits a known PBW basis. This comparison, rather than an independent basis for ^iY itself, establishes that no nontrivial kernel can exist. While we do not construct a new faithful representation or write out a PBW basis for ^iY in the present text, the embedding into Y together with the filtration argument suffices for the claim. We are nevertheless willing to add a short clarifying paragraph spelling out this graded comparison if the referee considers it helpful for readability. revision: partial

Circularity Check

0 steps flagged

No significant circularity: independent presentation and homomorphism supplied

full rationale

The paper defines a new minimalistic Drinfeld-type presentation for ^i Y (previously introduced in arXiv:2406.05067) and then derives an injective homomorphism into the ordinary Yangian Y together with coproduct formulas and generator estimates. These steps supply independent algebraic content rather than reducing by construction to the prior definition or to any fitted quantity. The reference to the earlier paper serves only as the definitional starting point for the object being presented; the homomorphism, coideal identification, and isomorphism to the J-presentation are established via explicit maps and relations that do not tautologically reproduce the inputs. No self-citation chain is load-bearing for the central claims, and the derivation remains self-contained against the external benchmark of the standard Yangian.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the prior definition of the twisted Yangian from arXiv:2406.05067 and on standard properties of Hopf algebra homomorphisms and coideals; no new free parameters or invented entities are introduced.

axioms (2)

domain assumption The twisted Yangian satisfies the relations of the proposed minimalistic Drinfeld presentation
This is the defining step for introducing the new presentation.
standard math Standard axioms of associative algebras and Hopf algebra coproducts
Used throughout the homomorphism and coideal arguments.

pith-pipeline@v0.9.0 · 5419 in / 1271 out tokens · 47014 ms · 2026-05-18T00:04:00.602730+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem A (minimalistic presentation generated by h_{i,1}, b_{i,0}, b_{i,1} subject to (3.1)–(3.3) plus Serre relations); Theorem B (injective φ: ıY → Y with explicit images (1.1)–(1.2)); Theorem C (generator estimates hi(u) ≡ ξ_i(u)ξ_i(−u) mod Y_{Q+[[u^{-1}]]})
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Drinfeld presentation relations (2.27)–(2.33) and coproduct formulas (2.18)–(2.21)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

[1]

Brundan and A

[BK05] J. Brundan and A. Kleshchev,Parabolic presentations of the YangianY(gl n), Comm. Math. Phys.254(2005), no. 1, 191–220. [BK06] J. Brundan and A. Kleshchev,Shifted Yangians and finiteW-algebras, Adv. Math.200(2006), no. 1, 136–195. [BPT25] R. Bartlett, T. Prze´ zdziecki, and L. Tappeiner,GKLO representations of twisted Yangians in typeAIand quantizat...

work page arXiv 2005
[2]

[CGM14] H. Chen, N. Guay, and X. Ma,Twisted Yangians, twisted quantum loop algebras and affine Hecke algebras of typeBC, Trans. Amer. Math. Soc.366(2014), no. 5, 2517–2574. [Che84] I. Cherednik,Factorizing particles on a half line, and root systems, Teoret. Mat. Fiz.61(1984), no. 1, 35–44. [Dri85] V. G. Drinfeld,Hopf algebras and the quantum Yang-Baxter e...

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Gavarini,The quantum duality principle, Ann

[Gav02] F. Gavarini,The quantum duality principle, Ann. Inst. Fourier (Grenoble)52(2002), no. 3, 809–834. [GNW18] N. Guay, H. Nakajima, and C. Wendlandt,Coproduct for Yangians of affine Kac-Moody algebras, Adv. Math.338(2018), 865–911. [GR16] N. Guay and V. Regelskis,Twisted Yangians for symmetric pairs of types B, C, D, Math. Z.284(2016), no. 1-2, 131–16...

work page arXiv 2002
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Hernandez and H

[HZ24] D. Hernandez and H. Zhang,Shifted Yangians and polynomialR-matrices, Publ. Res. Inst. Math. Sci.60(2024), no. 1, 1–69. [JLM18] N. Jing, M. Liu, and A. Molev,Isomorphism between theR-matrix and Drinfeld presentations of Yangian in typesB,C andD, Comm. Math. Phys.361(2018), no. 3, 827–872. [Kni95] H. Knight,Spectra of tensor products of finite-dimens...

work page 2024
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Lu, Y.-N

[LPT+25] K. Lu, Y.-N. Peng, L. Tappeiner, L. Topley, and W. Wang,Shifted twisted Yangians and finite W-algebras of classical type, arXiv preprint arXiv:2505.03316(2025). [Lu24] K. Lu,Isomorphism Between Twisted q-Yangians and AffineıQuantum Groups: Type AI, Int. Math. Res. Not. IMRN 2024(2024), no. 24, 14519–14542. MINIMAL PRESENTATION AND COIDEAL STRUCTU...

work page arXiv 2025
[6]

[LWW25] K. Lu, W. Wang, and A. Weekes,Shifted twisted Yangians and affine Grassmannian islices, arXiv preprint arXiv:2510. 10652(2025). [LWZ24] M. Lu, W. Wang, and W. Zhang,Braid group action and quasi-split affineıquantum groups II: Higher rank, Comm. Math. Phys.405(2024), no. 6, Paper No. 142,

work page 2025
[7]

[LWZ25a] K. Lu, W. Wang, and W. Zhang,A Drinfeld type Presentation of twisted Yangians, Represent. Theory29(2025), 838–870. [LWZ25b] K. Lu, W. Wang, and W. Zhang,Affineıquantum groups and twisted Yangians in Drinfeld presentations, Comm. Math. Phys.406(2025), no. 5, Paper No. 98,

work page 2025
[8]

Preprint;arXiv:2510.13007

[LZ24] K. Lu and W. Zhang,A Drinfeld type Presentation of twisted Yangians of quasi-split type, arXiv preprint arXiv:2408. 06981(2024). [Mac02] N. MacKay,RationalK-matrices and representations of twisted Yangians, J. Phys. A35(2002), no. 37, 7865–7876. [Mac05] N. J. MacKay,Introduction to Yangian symmetry in integrable field theory, Internat. J. Modern Ph...

work page arXiv 2024
[9]

Regelskis,Bethe vectors and recurrence relations for twisted Yangian based models, SciPost Phys.17(2024), no

[Reg24] V. Regelskis,Bethe vectors and recurrence relations for twisted Yangian based models, SciPost Phys.17(2024), no. 5, Paper No. 126,

work page 2024
[10]

Sklyanin,Boundary conditions for integrable quantum systems, J

[Skl88] E. Sklyanin,Boundary conditions for integrable quantum systems, J. Phys. A21(1988), no. 10, 2375–2389. [SSX25] Y. Shen, C. Su, and R. Xiong,Quivers with involutions and shifted twisted Yangians via Coulomb branches, arXiv preprint arXiv:2510.12118(2025). [TT24] L. Tappeiner and L. Topley,Shifted twisted Yangians and Slodowy slices in classical Lie...

work page arXiv 1988
[11]

Zhang,A Drinfeld type presentation of affineıquantum groups II: split BCFG type, Lett

[Zha22] W. Zhang,A Drinfeld type presentation of affineıquantum groups II: split BCFG type, Lett. Math. Phys.112(2022), no. 5, Paper No. 89,

work page 2022

[1] [1]

Brundan and A

[BK05] J. Brundan and A. Kleshchev,Parabolic presentations of the YangianY(gl n), Comm. Math. Phys.254(2005), no. 1, 191–220. [BK06] J. Brundan and A. Kleshchev,Shifted Yangians and finiteW-algebras, Adv. Math.200(2006), no. 1, 136–195. [BPT25] R. Bartlett, T. Prze´ zdziecki, and L. Tappeiner,GKLO representations of twisted Yangians in typeAIand quantizat...

work page arXiv 2005

[2] [2]

[CGM14] H. Chen, N. Guay, and X. Ma,Twisted Yangians, twisted quantum loop algebras and affine Hecke algebras of typeBC, Trans. Amer. Math. Soc.366(2014), no. 5, 2517–2574. [Che84] I. Cherednik,Factorizing particles on a half line, and root systems, Teoret. Mat. Fiz.61(1984), no. 1, 35–44. [Dri85] V. G. Drinfeld,Hopf algebras and the quantum Yang-Baxter e...

work page 2014

[3] [3]

Gavarini,The quantum duality principle, Ann

[Gav02] F. Gavarini,The quantum duality principle, Ann. Inst. Fourier (Grenoble)52(2002), no. 3, 809–834. [GNW18] N. Guay, H. Nakajima, and C. Wendlandt,Coproduct for Yangians of affine Kac-Moody algebras, Adv. Math.338(2018), 865–911. [GR16] N. Guay and V. Regelskis,Twisted Yangians for symmetric pairs of types B, C, D, Math. Z.284(2016), no. 1-2, 131–16...

work page arXiv 2002

[4] [4]

Hernandez and H

[HZ24] D. Hernandez and H. Zhang,Shifted Yangians and polynomialR-matrices, Publ. Res. Inst. Math. Sci.60(2024), no. 1, 1–69. [JLM18] N. Jing, M. Liu, and A. Molev,Isomorphism between theR-matrix and Drinfeld presentations of Yangian in typesB,C andD, Comm. Math. Phys.361(2018), no. 3, 827–872. [Kni95] H. Knight,Spectra of tensor products of finite-dimens...

work page 2024

[5] [5]

Lu, Y.-N

[LPT+25] K. Lu, Y.-N. Peng, L. Tappeiner, L. Topley, and W. Wang,Shifted twisted Yangians and finite W-algebras of classical type, arXiv preprint arXiv:2505.03316(2025). [Lu24] K. Lu,Isomorphism Between Twisted q-Yangians and AffineıQuantum Groups: Type AI, Int. Math. Res. Not. IMRN 2024(2024), no. 24, 14519–14542. MINIMAL PRESENTATION AND COIDEAL STRUCTU...

work page arXiv 2025

[6] [6]

[LWW25] K. Lu, W. Wang, and A. Weekes,Shifted twisted Yangians and affine Grassmannian islices, arXiv preprint arXiv:2510. 10652(2025). [LWZ24] M. Lu, W. Wang, and W. Zhang,Braid group action and quasi-split affineıquantum groups II: Higher rank, Comm. Math. Phys.405(2024), no. 6, Paper No. 142,

work page 2025

[7] [7]

[LWZ25a] K. Lu, W. Wang, and W. Zhang,A Drinfeld type Presentation of twisted Yangians, Represent. Theory29(2025), 838–870. [LWZ25b] K. Lu, W. Wang, and W. Zhang,Affineıquantum groups and twisted Yangians in Drinfeld presentations, Comm. Math. Phys.406(2025), no. 5, Paper No. 98,

work page 2025

[8] [8]

Preprint;arXiv:2510.13007

[LZ24] K. Lu and W. Zhang,A Drinfeld type Presentation of twisted Yangians of quasi-split type, arXiv preprint arXiv:2408. 06981(2024). [Mac02] N. MacKay,RationalK-matrices and representations of twisted Yangians, J. Phys. A35(2002), no. 37, 7865–7876. [Mac05] N. J. MacKay,Introduction to Yangian symmetry in integrable field theory, Internat. J. Modern Ph...

work page arXiv 2024

[9] [9]

Regelskis,Bethe vectors and recurrence relations for twisted Yangian based models, SciPost Phys.17(2024), no

[Reg24] V. Regelskis,Bethe vectors and recurrence relations for twisted Yangian based models, SciPost Phys.17(2024), no. 5, Paper No. 126,

work page 2024

[10] [10]

Sklyanin,Boundary conditions for integrable quantum systems, J

[Skl88] E. Sklyanin,Boundary conditions for integrable quantum systems, J. Phys. A21(1988), no. 10, 2375–2389. [SSX25] Y. Shen, C. Su, and R. Xiong,Quivers with involutions and shifted twisted Yangians via Coulomb branches, arXiv preprint arXiv:2510.12118(2025). [TT24] L. Tappeiner and L. Topley,Shifted twisted Yangians and Slodowy slices in classical Lie...

work page arXiv 1988

[11] [11]

Zhang,A Drinfeld type presentation of affineıquantum groups II: split BCFG type, Lett

[Zha22] W. Zhang,A Drinfeld type presentation of affineıquantum groups II: split BCFG type, Lett. Math. Phys.112(2022), no. 5, Paper No. 89,

work page 2022