Pith.

Quantum Algebra · arXiv:2610.30521 v1 · submitted 21 Oct 2026 · 76 references

Drinfeld presentation of affine iquantum groups via iHopf algebras

Ming Lu, Xiaolong Pan

PDF · arXiv · The abstract, sentence by sentence · Cite

The paper says

“Affine quantum groups $\widetilde{\mathbf{U}}$ and affine iquantum groups $\widetilde{\mathbf {U}}^\imath$ admit two complementary descriptions: a Serre-type presentation and a Drinfeld (current) presentation.”

more of the abstract

While iquantum groups $\widetilde{\mathbf {U}}^\imath$ are typically defined via the Serre presentation, their Drinfeld presentations are substantially more subtle and less well understood. In this paper, we give a conceptual derivation of the Drinfeld presentation of $\widetilde{\mathbf {U}}^\imath$ starting directly from the Drinfeld presentation of $\widetilde{\mathbf {U}}$, working within the framework of iHopf algebras. This approach leads naturally to the definition of iquantum affinization associated with arbitrary Kac-Moody algebras $\mathfrak{g}$. We further show that, in this general setting, these iquantum affinizations can be realized as coideal subalgebras of quantum affinizations with respect to the Drinfeld coproduct. Finally, in the affine case, we describe the imaginary root vectors Θi,r of $\widetilde{\mathbf {U}}^\imath$ explicitly in terms of the root vectors ψ±i,r of $\widetilde{\mathbf {U}}$, thereby extending the results of Li and Przeździecki for certain affine types of $\widetilde{\mathbf {U}}^\imath$.

from the abstract

Pith says

Machine review · deepseek-v4-flash · 23 Oct 2026 · what was read was not recorded · not checked by a person

It can be wrong. Challenge this review or reply to it.

The passes disagreed. Reader: holds up. Skeptic: holds up only with conditions. Referee: major revision.

A genuinely new conceptual approach that unifies Drinfeld presentations for affine iquantum groups via iHopf algebras—significant advance, but Lemma 4.11's verification needs tightening. This line is wrong

2 major objections · 3 minor

How often objections from reviews like this one are right: not yet measured

Are you an author? What you can do.

Not checked: figures, code, what the 76 references say.

The paper says, at §4.3, Lemma 4.11

The review did not quote this passage.

Pith says: Objection 1 of 2 · major

The verification that the τ-twisted compatible linear map Drχ descends to the quotient by the Serre relations (4.16) is not rigorous. The argument sketched after the line 'It remains to verify that Drχ vanishes on the ideal generated by the Serre relation (4.16)' relies on weight arguments and an unsubstantiated claim that 'Drχ(a)=0 if r>0 from the definition of Drχ (and similar for Drχ(c))'.

more of objection 1

Because Drχ is defined via the τ-twisted compatibility condition (which itself involves infinite coproduct sums in the topological setting), the behaviour of Drχ on a product containing a θ_i generator is not obviously zero. The coproduct Δ(θ_i(z)) contains an infinite series ψ_i(z)⊗θ_i(zC_1) and the pairing involves infinite sums; the argument does not address how these infinite series are handled or why the stated vanishing holds. Since Lemma 4.11 is essential for the embedding Dr eξ_τ (Proposition 4.12) and consequently for the isomorphism in Theorem 4.7 and the embedding in Theorem 4.13, this gap is load-bearing. The authors should provide a complete proof that Drχ vanishes on the Serre ideal, or at least explain why the infinite coproduct contributions can be controlled (e.g., by the Z_{I0}-grading and the topology of the completed algebra).

The paper says, at §4.3, Lemma 4.11

The review did not quote this passage.

Pith says: Objection 2 of 2 · major

The definition of Drχ and the verification that it satisfies the τ-twisted compatibility condition rely on the weak Hopf pairing ⟨−,−⟩. The pairing is defined on Dr eB, but Dr eB is a topological algebra; the sums in the τ-twisted compatibility condition are not guaranteed to be finite.

more of objection 2

The paper does not discuss the topology or the sense in which these sums converge. A rigorous treatment should either show that only finitely many terms contribute (using gradings) or specify a completion with respect to which the sums converge. Without this, the existence of the map Drχ and its compatibility are not fully established.

3 minor objections

The paper says, at §3.3, Eq. (3.15)–(3.16)

Pith says: minor objection 1 of 3

The coproduct formulas for E_{mδ,i} and E_{rδ+α_i} are stated as in [Da24]. A brief explanation of the notation eU^{≥0}(+∞), eU^+(−∞), etc., would improve readability, as these symbols are used in the later coproduct expansion arguments.

The paper says, at §4.3, Eq. (4.28)–(4.29)

Pith says: minor objection 2 of 3

The definition of the idivided powers B_{i,l}^{(m);\overline{1}} and B_{i,l}^{(m);\overline{0}} is given but the indices \overline{1},\overline{0} are not explicitly linked to the parity or to the parameter p_i in (4.32). Adding a short explanation would be helpful.

show 1 more

The paper says, at §4.3, Eq. (4.50)–(4.54)

Pith says: minor objection 3 of 3

The explicit formulas for the iSerre relations in the general case (c_{ij} c_{ji} ≤ 3) are presented without derivation. While these are consequences of the main theorem once proven, the paper states 'we omit the details'. At least a sketch of the derivation or a reference to the analogous computation in [L W21, Z22] would improve the exposition.

What would settle it

Verify Lemma 4.11 explicitly for g = sl_3 (i=1, j=2, τ=Id, c_{12}=-1). Compute Drχ on each coefficient of the power series expansion of the Serre relation (4.16) for this case, using the explicit formulas for Drχ(ψ_i(u)) = (1-u^2)/(1-v_i^{-2}u^2) and the coproduct (4.17)–(4.18) with the full infinite sum. Check that the sum of terms vanishes identically. If any coefficient is nonzero, Lemma 4.11 fails and the central construction is unsupported.

from the skeptic's reading

How the review went

4 readings by deepseek-v4-flash · 23 Oct 2026

  1. Reader holds up confidence high
  2. Skeptic agrees: says it holds up only with conditions 46 s · 42,541 tokens read, 4,652 written Lemma 4.11's verification that the τ-twisted compatible map Drχ descends to Dr eB is incomplete; the argument for vanishing on the Serre relations relies on weight arguments that are not fully justified, especially for the infinite coproduct sums.
  3. Editor wrote the verdict above 27 s · 43,421 tokens read, 2,513 written
  4. Referee major revision: 2 major objections, 3 minor 55 s · 42,980 tokens read, 4,473 written

Not run on this paper: the simulated authors' defence.

Pith did not check this review's objections against the paper.

The review in full

The review in brief, machine-written

This paper does something that needed doing. The iHopf algebra method replaces a stack of case-by-case calculations (split ADE, BCD, AIII, etc.) with a single conceptual derivation of the Drinfeld presentation for affine iquantum groups. It also defines iquantum affinization for arbitrary Kac-Moody algebras and embeds it into the quantum affinization as a coideal subalgebra. Theorems A, C, and D are clean and plausible. The framework is the real contribution: once you have the iHopf machinery (from the companion paper [CLPRW25]), the results essentially fall out of the Hopf algebra structure of the Borel subalgebra and its Drinfeld version.

The soft spot is Lemma 4.11, the construction of the τ-twisted compatible map Drχ on the Drinfeld Borel subalgebra. The proof that Drχ descends to the quotient by the Serre relations is sketched rather than fully justified. The argument uses weight-gradedness of the weak Hopf pairing and a reduction to the generator u_ij, but the expansion of the coproduct in modes involves partial sums over ψi,s that deserve a more explicit verification. The claim that 'we conclude that Drχ(abc) = 0 unless cij=0' and the final handling of that case rely on weight constraints that are plausible but not spelled out. A referee will want this step tightened.

That said, this is a technical gap in one lemma, not a flaw in the central idea. The rest of the paper's logic—especially Theorem 4.7, which also uses a separate argument via the finite-type embedding to handle Serre relations—does not depend entirely on that lemma. The paper is clearly written by experts, and the overall picture is convincing.

This paper is for specialists in quantum symmetric pairs, Drinfeld presentations, and Hopf algebras. It deserves a serious referee and should be accepted after the authors clarify Lemma 4.11.

The strongest objection

The entire construction of Dr eU^ı and the isomorphism Theorem 4.7 hinge on Lemma 4.11, which defines a τ-twisted compatible linear map Drχ on the Drinfeld Borel subalgebra Dr eB. The proof that Drχ descends to the quotient by the Serre relations (4.16) is sketched but not rigorous. The argument uses the Z_{I0}-gradedness of the weak Hopf pairing and the fact that Drχ(θ_{i,r})=0 to claim that most terms in the expansion of Drχ on the Serre combination vanish, but it does not handle the infinite sums in the coproduct Δ(θ_i(z)) = θ_i(z)⊗1 + h_i ψ_i(z) ⊗ θ_i(zC_1) + ... (which includes an infinite series over ψ_{i,l} and θ_{i,-l} in the full Dr eB). The proof that Drχ(u_ij)=0 for the generator u_ij relies on weight constraints that are plausible but not explicitly verified for the series contributions. Moreover, the reliance on Lemma 4.4 to reduce to the generator is not explained. If Drχ does not vanish on the Serre ideal, then the embedding Dr eξ_τ (Proposition 4.12) and consequently Theorems 4.7 and 4.13 are not valid. Given the centrality of this step, the gap is load-bearing.

The reader's weakest assumption correctly identifies that the iHopf algebra construction and the τ-twisted compatible map are load-bearing. The paper's Lemma 4.11 is the critical point where the existence of this map on the Drinfeld Borel subalgebra is claimed, and its proof is not fully rigorous. The argument for the Serre relations is sketchy and does not satisfactorily handle the infinite coproduct expansions that appear in the Drinfeld presentation. A concrete check for a simple case would resolve whether the proof is correct or needs additional work. Therefore, the verdict should be CONDITIONAL on a more detailed verification or an independent computational check.

The referee's report

If the main results hold, the paper provides a significant conceptual unification: the Drinfeld presentation of affine iquantum groups, previously established only in a series of case-by-case works, is derived from the quantum group Drinfeld presentation via the iHopf algebra construction. This approach simplifies the derivation and makes the underlying algebraic structure transparent. The extension to iquantum affinizations for arbitrary Kac-Moody algebras is a new construction that generalises affine iquantum groups and is likely to be influential in the representation theory of quantum symmetric pairs and related areas (e.g., boundary q-characters, quantum integrable systems). The explicit identification of the imaginary root vectors of the iquantum group with certain rational expressions in the quantum affine root vectors is an elegant and useful formula.

Pith's reading of the paper (a machine's summary, not the authors')

This paper uses the iHopf algebra framework to give a single, conceptual derivation of the Drinfeld (current) presentation of affine iquantum groups $\widetilde{\mathbf{U}}^{\imath}$ from the Drinfeld presentation of the affine quantum group $\widetilde{\mathbf{U}}$. The authors construct an algebra isomorphism $\Xi : \operatorname{Dr}\mathbf{B}^{\imath}_{\tau} \to \operatorname{Dr}\mathbf{U}^{\imath}$ that identifies the iHopf algebra generators with the Drinfeld generators of the iquantum loop algebra. This approach unifies previous case-by-case constructions and extends naturally to arbitrary Kac-Moody algebras, defining iquantum affinizations as coideal subalgebras of quantum affinizations.

What carries the argument. The key machinery is the iHopf algebra construction of Chen–Lu–Pan–Ruan–Wang [CLPRW25]: given a Hopf algebra $H$ with a Hopf pairing $\varphi$, one defines a new associative algebra $H^{\imath}_{\tau}$ on the same vector space by twisting the multiplication with a $\tau$-twisted compatible map $\chi$. Applied to the Drinfeld Borel subalgebra $\operatorname{Dr}\mathbf{B}$ (a topological Hopf algebra with a weak Hopf pairing $\langle\cdot,\cdot\rangle$) and a $\tau$-twisted compatible map $\operatorname{Dr}\chi$, this produces $\operatorname{Dr}\mathbf{B}^{\imath}_{\tau}$. The isomorphism $\Xi$ then identifies $\operatorname{Dr}\mathbf{B}^{\imath}_{\tau}$ with the iquantum loop algebra $\operatorname{Dr}\mathbf{U}^{\imath}$, and the embedding $L_{\operatorname{Dr}}$ realises it as a subalgebra of $\operatorname{Dr}\mathbf{U}$.

Core claim. The central discovery is that the Drinfeld presentation of affine iquantum groups can be obtained uniformly by applying the iHopf algebra construction to the Drinfeld Borel subalgebra $\operatorname{Dr}\mathbf{B}$ of the quantum affinization. Specifically, for any involution $\tau$ of the Cartan matrix that fixes the affine node, there is an algebra isomorphism $\Xi: \operatorname{Dr}\mathbf{B}^{\imath}_{\tau} \to \operatorname{Dr}\mathbf{U}^{\imath}$ mapping $H_i \mapsto K_{\tau i}$, $\theta_{i,r} \mapsto B_{i,r}$, $\psi_{i,m} \mapsto \Theta_{i,m}$, and $C \mapsto C$. This shows that the generators $B_{i,r}, \Theta_{i,m}$ of the iquantum group coincide with the images of the Borel root vectors under a twisted multiplication. The paper also provides an explicit embedding $L_{\operatorname{Dr}}: \operatorname{Dr}\mathbf{U}^{\imath} \hookrightarrow \operatorname{Dr}\mathbf{U}^T$ realizing the iquantum loop algebra as a coideal subalgebra of the quantum loop algebra.

The premise it rests on. The construction assumes that the weak Hopf pairing on the Drinfeld Borel subalgebra $\operatorname{Dr}\mathbf{B}$ satisfies the necessary identities and that the $\tau$-twisted compatible map $\operatorname{Dr}\chi$ defined in Lemma 4.11 respects all defining relations of $\operatorname{Dr}\mathbf{B}$, including the Serre relations, which is verified through detailed calculations.

If the paper is right (Pith's inferences, not the authors')

All previously established Drinfeld presentations of affine iquantum groups (split ADE, BCFG, quasi-split types) are now seen as special cases of a single isomorphism, significantly simplifying their derivation.

The construction defines iquantum affinization for arbitrary Kac-Moody algebras, not just affine ones, and proves it is a coideal subalgebra of the corresponding quantum affinization.

The explicit description of the imaginary root vectors $\Theta_{i,m}$ in terms of $\psi^{\pm}_{i,\pm m}$ (Theorem 3.18) extends earlier results in classical types to all untwisted affine iquantum groups, forming the basis for boundary $q$-characters.

The results suggest that analogous Drinfeld-type presentations for twisted affine iquantum groups and twisted Yangians can be obtained from the same iHopf algebra framework.

The iHopf algebra method should apply to twisted affine types without new conceptual difficulty, reducing those constructions to the known Drinfeld presentation of twisted quantum affine algebras.

The degeneration of affine iquantum groups to twisted Yangians is expected to yield a unified Drinfeld presentation of twisted Yangians, replacing the current case-by-case Gauss decomposition computations.

The iquantum affinization for arbitrary Kac-Moody algebras may provide a new family of quantum symmetric pairs that are coideal subalgebras of quantum affinizations, with potential applications to Hall algebras and geometric representation theory.

Check this review

Pith stored no record of this review's passes and took no fingerprint of it. Reviews made by code from 13 Nov 2026 on carry both.

How often reviews like this one are right

Reviews like this one: written by deepseek-v4-flash, where what was read was not recorded.

How often objections from reviews like this one are right: not yet measured

Every value above, in the review's public record (JSON)

Reviews by people

This review was written by a machine. No person has reviewed this paper on Pith yet. A scientist listed on Pith can review it for a fee, and their review is published here whatever it concludes.

Request a human review

How Pith checks papers · The full audit of this paper

The paper

Abstract

Affine quantum groups $\widetilde{\mathbf{U}}$ and affine iquantum groups $\widetilde{\mathbf {U}}^\imath$ admit two complementary descriptions: a Serre-type presentation and a Drinfeld (current) presentation. While iquantum groups $\widetilde{\mathbf {U}}^\imath$ are typically defined via the Serre presentation, their Drinfeld presentations are substantially more subtle and less well understood. In this paper, we give a conceptual derivation of the Drinfeld presentation of $\widetilde{\mathbf {U}}^\imath$ starting directly from the Drinfeld presentation of $\widetilde{\mathbf {U}}$, working within the framework of iHopf algebras. This approach leads naturally to the definition of iquantum affinization associated with arbitrary Kac-Moody algebras $\mathfrak{g}$. We further show that, in this general setting, these iquantum affinizations can be realized as coideal subalgebras of quantum affinizations with respect to the Drinfeld coproduct. Finally, in the affine case, we describe the imaginary root vectors Θi,r of $\widetilde{\mathbf {U}}^\imath$ explicitly in terms of the root vectors ψ±i,r of $\widetilde{\mathbf {U}}$, thereby extending the results of Li and Przeździecki for certain affine types of $\widetilde{\mathbf {U}}^\imath$.

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. No sentence in this record was written by a model.

abstract compiled · 2026-10-23

The authors' main claim, as the classifier picked it out 0.89

“In this paper, we give a conceptual derivation of the Drinfeld presentation of $\widetilde{\mathbf {U}}^\imath$ starting directly from the Drinfeld presentation of $\widetilde{\mathbf {U}}$, working within the framework of iHopf algebras.”

How the main claim is stated: asserted 1.00

What the abstract itself states, as probabilities: claims first 0.08 · states limitation 0.17 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.02 · compares with prior results 0.90 · 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.

SentenceRoleThe authors' own workStates the main conclusionStrength
Affine quantum groups $\widetilde{\mathbf{U}}$ and affine iquantum groups $\widetilde{\mathbf {U}}^\imath$ admit two complementary descriptions: a Serre-type presentation and a Drinfeld (current) presentation. background 1.00 0.08 0.05 asserted 1.00
While iquantum groups $\widetilde{\mathbf {U}}^\imath$ are typically defined via the Serre presentation, their Drinfeld presentations are substantially more subtle and less well understood. gap 0.83 0.04 0.03 asserted 0.98
In this paper, we give a conceptual derivation of the Drinfeld presentation of $\widetilde{\mathbf {U}}^\imath$ starting directly from the Drinfeld presentation of $\widetilde{\mathbf {U}}$, working within the framework of iHopf algebras. contribution 0.87 0.97 0.61 asserted 1.00
This approach leads naturally to the definition of iquantum affinization associated with arbitrary Kac-Moody algebras $\mathfrak{g}$. result 0.47 0.86 0.14 asserted 0.99
We further show that, in this general setting, these iquantum affinizations can be realized as coideal subalgebras of quantum affinizations with respect to the Drinfeld coproduct. result 0.94 0.94 0.18 asserted 1.00
Finally, in the affine case, we describe the imaginary root vectors $\Theta_{i,r}$ of $\widetilde{\mathbf {U}}^\imath$ explicitly in terms of the root vectors $\psi^\pm_{i,r}$ of $\widetilde{\mathbf {U}}$, thereby extending the results of Li and Prze\'zdziecki for certain affine types of $\widetilde{\mathbf {U}}^\imath$. result 0.79 0.95 0.19 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 1.00

What was examined, and what was not

Sources read

  • arXiv listing abstract v1

Not examined

  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers

Levels run

  • P abstract

Not run

  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

How the labels were made

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-23: 1 request, 5,349 input tokens.

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. This record as JSON

References (76)
  1. [1] A. Appel, T. Prze\'zdziecki, Generalized Schur-Weyl dualities for quantum affine symmetric pairs and orientifold KLR algebras , Adv. Math. 435 (2023), Paper No. 109383 on Pith
  2. [2] A. Appel and B. Vlaar, Universal K-matrices for quantum Kac-Moody algebras , Represent. Theory 26 (2022), 764--824 on Pith
  3. [3] M. Balagovic and S. Kolb, Universal K-matrix for quantum symmetric pairs , J. Reine Angew. Math. 747 (2019), 299--353 on Pith
  4. [4] H. Bao and W. Wang, A new approach to Kazhdan-Lusztig theory of type B via quantum symmetric pairs , Ast\'erisque 402 , 2018, vii+134pp on Pith
  5. [5] H. Bao and W. Wang, Canonical bases arising from quantum symmetric pairs , Inventiones Math. 213 (2018), 1099--1177 on Pith
  6. [6] H. Bao and W. Wang, Canonical bases arising from quantum symmetric pairs of Kac-Moody type , Compositio Math. 157 (2021), 1507–1537 on Pith
  7. [7] P. Baseilhac, S. Belliard, Generalized q-Onsager algebras and boundary affine Toda field theories , Lett. Math. Phys. 93 (2010), 213--228 on Pith
  8. [8] J. Beck, Braid group actions and quantum affine algebras , Commun. Math. Phys. 165 (1994), 555--568 on Pith
  9. [9] A. Berenstein and J. Greenstein, Double canonical bases , Adv. Math. 316 (2017), 381--468 on Pith
  10. [10] A. Berenstein and J. Greenstein, Canonical bases of quantum Schubert cells and their symmetries , Sel. Math. New Ser. 23, 2755–2799 (2017) on Pith
  11. [11] A. Berenstein and A. Zelevinsky, Triangular Bases in Quantum Cluster Algebras , IMRN 2014 (2014), no. 6, 1651--1688 on Pith
  12. [12] C. Berman and W. Wang, Formulae of -divided powers in U _q( sl _2) , J. Pure Appl. Algebra 222 (2018), 2667--2702 on Pith
  13. [13] V. Chari and D. Hernandez, Beyond Kirillov-Reshetikhin modules, Contemp. Math. 506 , 49-81, Amer. Math. Soc., Providence, RI (2010) on Pith
  14. [14] J. Chen, M. Lu, X. Pan, S. Ruan and W. Wang, i Quantum groups and i Hopf algebras I: foundations , arXiv:2511.11291 https://arxiv.org/abs/2511.11291 on PithPith's reviewarXiv
  15. [15] J. Chen, M. Lu, X. Pan, S. Ruan and W. Wang, i Quantum groups and i Hopf algebras II: dual canonical bases , arXiv:2601.00524 https://arxiv.org/abs/2601.00524 on PitharXiv
  16. [16] X. Chen, M. Lu and W. Wang, A Serre presentation for the quantum groups , Transform. Groups 26 (2021), 827--857 on Pith
  17. [17] X. Chen, M. Lu and W. Wang, on Pith
  18. [18] T. Cramer, Double Hall algebras and derived equivalences , Adv. Math. 224 (2010), 1097--1120 on Pith
  19. [19] I. Damiani, La R-matrice pour les alg\`ebres quantiques de type affine non tordu , Ann. Sci. \'Ec. Norm. Sup\'er. (4) 31 (1998), 493–523 on Pith
  20. [20] I. Damiani, Drinfeld realization of affine quantum algebras: the relations , Publ. Res. Inst. Math. Sci. 48 (2012), 661--733 on Pith
show 56 more
  1. [21] I. Damiani, From the Drinfeld realization to the Drinfeld-Jimbo presentation of affine quantum algebras: injectivity , Publ. Res. Inst. Math. Sci. 51 (2015), 131--171 on Pith
  2. [22] I. Damiani, On the Drinfeld coproduct , Pure Appl. Math. Q. 20 (2024), no. 1, 171--232 on Pith
  3. [23] L. Dobson and S. Kolb, Factorisation of quasi K-matrices for quantum symmetric pairs , Selecta Math. (N.S.) 25 (2019), Paper No. 63, 53 pp on Pith
  4. [24] V.G. Drinfeld, Quantum groups , in: Proceedings of the International Congress of Mathematicians, vols. 1, 2, Berkeley, Calif., 1986, Amer. Math. Soc., Providence, RI, 1987, pp. 798–820 on Pith
  5. [25] V.G. Drinfeld, A new realization of Yangians and quantum affine algebras , Soviet. Math. Dokl. 36 (2)(1988), 212--216 on Pith
  6. [26] B. Enriquez, S. Khoroshkin, S. Pakuliak, Weight functions and Drinfeld currents , Comm. Math. Phys. 276 (2007), 691--725 on Pith
  7. [27] Z. Fan, C. Lai, Y. Li, L. Luo, W. Wang, Affine flag varieties and quantum symmetric pairs , Mem. Amer. Math. Soc. 265 (2020), No. 1285 on Pith
  8. [28] Z. Fan, C. Lai, Y. Li, L. Luo, W. Wang, H. Watanabe, Quantum Schur duality of affine type C with three parameters , Math. Res. Lett. 27 (2020), 79--114 on Pith
  9. [29] E. Frenkel, N. Reshetikhin, The q-characters of representations of quantum affine algebras and deformations of W-algebras , in: Recent Developments in Quantum Affine Algebras and Related Topics, Contemp. Math. 248 (1999), 163--205 on Pith
  10. [30] V. Ginzburg, M. Kapranov and E. Vasserot, Langlands reciprocity for algebraic surfaces , Math. Res. Lett. 2 (1995), 147--160 on Pith
  11. [31] P. Gross\' e , On quantum shuffle and quantum affine algebras, J. Algebra 318 (2007), 495--519 on Pith
  12. [32] D. Hernandez, Representations of quantum affinizations and fusion product , Transform. Groups 10 (2005), 163--200 on Pith
  13. [33] D. Hernandez, Drinfeld coproduct, quantum fusion tensor category and applications , Proc. Lond. Math. Soc. 95 (2007), 567--608 on Pith
  14. [34] D. Hernandez, Quantum toroidal algebras and their representations , Sel. Math. New Ser. 14 (3-4) (2009), 701--725 on Pith
  15. [35] D. Hernandez, Representations and characters of quantum affine algebras at the crossroads between cluster categorification and quantum integrable models , arXiv:2510.0643 https://arxiv.org/abs/2510.06437 on PitharXiv
  16. [36] D. Hernandez and B. Leclerc, Quantum Grothendieck rings and derived Hall algebras , J. Reine Angew. Math. 701 (2015), 77--126 on Pith
  17. [37] J.C. Jantzen, Lectures on quantum groups, Graduate studies in mathematics, Vol. 6 , AMS, 1995 on Pith
  18. [38] M. Jimbo, A q -Difference analogue of U( g) and the Yang-Baxter equation , Lett. Math. Phys. 10 (1985), 63--69 on Pith
  19. [39] N. Jing, Quantum Kac-Moody algebras and Vertex representations , Lett. Math. Phys. 44 (1998), 261--271 on Pith
  20. [40] A. Joseph, Quantum Groups and Their Primitive Ideals, Ergebnisse der Math. (3), vol. 29 , Springer-Verlag, Berlin, 1995 on Pith
  21. [41] S. Kato, Poincar\' e -Birkhoff-Witt bases and Khovanov-Lauda-Rouquier algebras , Duke Math. J. 163 (2014), 619--663 on Pith
  22. [42] S. Kolb, Quantum symmetric Kac-Moody pairs , Adv. Math. 267 (2014), 395--469 on Pith
  23. [43] S. Kolb and J. Pellegrini, Braid group actions on coideal subalgebras of quantized enveloping algebras , J. Algebra 336 (2011), 395--416 on Pith
  24. [44] S. Kolb and M. Yakimov, Symmetric pairs for Nichols algebras of diagonal on Pith
  25. [45] S. Kolb and M. Yakimov, Defining relations of quantum symmetric pair coideal on Pith
  26. [46] G. Letzter, Symmetric pairs for quantized enveloping algebras , J. Algebra 220 (1999), 729--767 on Pith
  27. [47] J.-R. Li, T. Prze\'zdziecki, Compatibility of Drinfeld presentations for split affine Kac-Moody quantum symmetric pairs . Lett. Math. Phys. 115 (2025), Paper No. 75 on Pith
  28. [48] J.-R. Li, T. Prze\'zdziecki, Boundary q -characters of evaluation modules for split quantum on Pith
  29. [49] J.-R. Li, T. Prze\'zdziecki, Compatibility of Drinfeld presentations and q-characters for affine Kac-Moody quantum symmetric pairs: quasi-split case , arXiv:2601.02165 https://arxiv.org/abs/2601.02165 on PitharXiv
  30. [50] K. Lu, W. Wang, W. Zhang, A Drinfeld type presentation of twisted Yangians , Representation Theory 29 (2025), 838--870 on Pith
  31. [51] K. Lu, W. Wang, W. Zhang, Affine quantum groups and twisted Yangian in Drinfeld presentations , Comm. Math. Phys. 406 (2025), Paper No. 98 on Pith
  32. [52] K. Lu, W. Zhang, A Drinfeld type presentation of twisted Yangians of quasi-split type , Commun. Contemp. Math. (to appear), 2026 on Pith
  33. [53] J. Lu, On the Drinfeld double and the Heisenberg double of a Hopf algebra , Duke Math. J. 74 (3) (1994), 763-776 on Pith
  34. [54] M. Lu and X. Pan, Dual canonical bases of quantum groups and quantum groups , arXiv:2504.19073 https://arxiv.org/abs/2504.19073 on PitharXiv
  35. [55] M. Lu, X. Pan, W. Wang and W. Zhang, Braid group action and quasi-split affine iquantum groups III , arXiv:2511.00882 https://arxiv.org/abs/2511.00882 on PitharXiv
  36. [56] M. Lu and S. Ruan, Hall algebras of weighted projective lines and quantum symmetric pairs III: quasi-split type , arXiv:2411.13078 https://arxiv.org/abs/2411.13078 on PithPith's reviewarXiv
  37. [57] M. Lu and W. Wang, A Drinfeld type presentation of affine ıquantum groups I: split ADE type , Adv. in Math. 393 (2021), 108111, 46 pp on Pith
  38. [58] M. Lu and W. Wang, Hall algebras and quantum symmetric pairs III: quiver varieties , Adv. Math. 393 (2021), 108071 on Pith
  39. [59] M. Lu and W. Wang, Hall algebras and quantum symmetric pairs I: foundations , Proc. London Math. Soc. (3) 124 (2022), 1--82 on Pith
  40. [60] M. Lu and W. Wang, Braid group symmetries on quasi-split quantum groups via Hall algebras , Selecta Math.(N.S.) 28 (2022), Paper No. 84 on Pith
  41. [61] M. Lu and W. Wang, Hall algebras and quantum symmetric pairs of Kac-Moody type , Adv. Math. 430 (2023), 109215 on Pith
  42. [62] M. Lu, W. Wang and W. Zhang, Braid group action and quasi-split affine quantum groups I , Represent. Theory 27 (2023), 1000--1040 on Pith
  43. [63] M. Lu, W. Wang and W. Zhang, Braid group actions and quasi-split affine ıquantum groups II: higher ranks, Comm. Math. Phys. 405 (2024): 142 on Pith
  44. [64] G. Lusztig, Introduction to Quantum Groups, Birkh\" a user, Boston, 1993 on Pith
  45. [65] A. Molev, E. Ragoucy, P. Sorba, Coideal subalgebras in quantum affine algebras , Rev. Math. Phys. 15 (8) (2003), 789--822 on Pith
  46. [66] H. Nakajima, Quiver varieties and finite-dimensional representations of quantum affine algebras , J. Amer. Math. Soc. 14 (2001), 145--238 on Pith
  47. [67] T. Prze\'zdziecki, Drinfeld rational fractions for affine Kac-Moody quantum symmetric pairs , Selecta Math.(N.S.) 31 (2025), Paper No. 87 on Pith
  48. [68] D. Radford, Hopf algebras with projection , J. Algebra 92 (1985), 322--347 on Pith
  49. [69] C.M. Ringel, Hall algebras and quantum groups , Invent. Math. 101 (1990), 583--591 on Pith
  50. [70] Y. Shen, C. Su, and R. Xiong, Quivers with involutions and shifted twisted Yangians via Coulomb branches , arXiv:2510.12118 https://arxiv.org/abs/2510.12118 on PithPith's reviewarXiv
  51. [71] E. Sklyanin, Boundary conditions for integrable quantum systems , J. Phys. A 21 (2375)(1988), 2375--2389 on Pith
  52. [72] C. Su and W. Wang, Affine iquantum groups and Steinberg varieties of type C , Selecta Math.(N.S.) 32 (2026), no. 3, Paper No. 50 on Pith
  53. [73] M. Varagnolo and E. Vasserot, Schur duality in the toroidal setting , Comm. Math. Phys. 182 (1996), 469–483 on Pith
  54. [74] W. Wang and W. Zhang, An intrinsic approach to relative braid group symmetries on quantum groups , Proc. London Math. Soc. 127 (2023), 1338--1423 on Pith
  55. [75] W. Zhang, A Drinfeld type presentation of affine quantum groups II: split BCFG type , Lett. Math. Phys. 112 (2022), 89 on Pith
  56. [76] W. Zhang, Relative braid group symmetries on quantum groups of Kac-Moody type , Selecta Math.(N.S.) 29 (2023), Paper No. 59, 54 pp on Pith
Cited by 1 Pith paper

Reviewed papers on Pith that cite this one. The full citation record

  1. FRT presentation of the $q$-Onsager comodule algebra and refined $q$-characters math.QA · Nov 2026

Challenges and discussion

Disagree with the review, or know something it missed? Leave a comment, or challenge this review formally: a challenge becomes a permanent entry in this paper's record, and its decision is published here.

No comments yet.

Sign in to comment