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arxiv: 2512.21750 · v2 · submitted 2025-12-25 · 🧮 math.QA · math-ph· math.MP

Extensions of a commuting pair of quantum toroidal mathfrak{gl}₁

B. Feigin , M. Jimbo , E. Mukhin This is my paper

Pith reviewed 2026-05-16 19:44 UTC · model grok-4.3

classification 🧮 math.QA math-phmath.MP MSC 17B3717B65
keywords quantum toroidal algebragl_1commuting subalgebrascoproductFock modulesshifted quantum toroidal gl_2algebra extensionsDrinfeld coproduct
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The pith

Algebras A_{M,N} extend commuting quantum toroidal gl_1 pairs by tuning parameters to M and N with a conjectured coproduct.

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

The paper introduces a family of algebras called A_{M,N} for integers M and N. These algebras contain a pair of commuting quantum toroidal gl_1 subalgebras whose parameters are adjusted according to M and N. Special cases with M equal to plus or minus one recover the shifted quantum toroidal gl_2 algebras. The authors conjecture the existence of a coproduct homomorphism from A_{M,N1 plus N2} to the completed tensor product of A_{M,N1} and A_{M,N2} that restricts to the Drinfeld coproduct on the subalgebras. Concrete modules are constructed as direct sums of tensor products of Fock modules for the pair of subalgebras.

Core claim

We introduce a family of algebras A_{M,N}, M,N in Z, as an extension of a pair of commuting quantum toroidal gl_1 subalgebras E1 and check E1, wherein the parameters are tuned in a specific way according to M,N. In the case M=±1, algebra A_{±1,N} is a shifted quantum toroidal gl_2 algebra. Conjecturally there is a coproduct homomorphism A_{M,N1+N2} to A_{M,N1} hat tensor A_{M,N2} whose restriction to the subalgebras coincides with the standard Drinfeld coproduct. We give examples of A_{M,N} modules constructed on certain direct sums of tensor products of Fock modules of E1 tensor check E1.

What carries the argument

The family of algebras A_{M,N} extending the commuting pair of quantum toroidal gl_1 subalgebras with M and N dependent parameters in the relations to support the conjectured coproduct.

Where Pith is reading between the lines

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

  • If the coproduct conjecture holds it would allow inductive construction of modules for larger N by combining those of smaller N.
  • The Fock module constructions suggest these algebras act naturally on spaces that combine multiple independent Fock spaces in a controlled way.
  • The general M,N family provides a uniform setting that recovers known shifted gl_2 cases and may extend to other rank-one quantum toroidal structures.
  • Computational verification of the coproduct for concrete small M and N would give direct evidence supporting the parameter tuning.

Load-bearing premise

The parameters in the algebra relations can be chosen depending on M and N so that the two quantum toroidal gl_1 subalgebras commute and the map defined as the conjectured coproduct is a homomorphism.

What would settle it

An explicit check for small values such as M=2 and N=1 whether the proposed coproduct map from A_{M,N1+N2} preserves all relations when mapped into the completed tensor product of the two smaller algebras.

Figures

Figures reproduced from arXiv: 2512.21750 by B. Feigin, E. Mukhin, M. Jimbo.

Figure 1
Figure 1. Figure 1: Coassociativity. At this writing it is not clear to us whether the formulas in (iii) are well-defined and compati￾ble with the relations (R1)–(R3). We shall come back to this point when we discuss examples of representations of AM,N . 4. Representations. In this section we give examples of representations of AM,N . In all cases considered here, the representations are such that when restricted to KM,N they… view at source ↗
Figure 2
Figure 2. Figure 2: The gl(2n|1) Dynkin diagram and labeling. The currents e(z), f(z) give the qq-character corresponding to the (2n + 1) dimensional represen￾tation (s1 and s2 are switched from [FJM]) n n−1 . . . . . . 1 0 1 n A−1 n,s1 A −1 n−1,s2 1 A −1 2,s n−1 1 A −1 1,sn 1 A −1 1,sn 1 s2 A −1 2,s n+1 1 s2 A −1 n,s 2n−1 1 s2 [PITH_FULL_IMAGE:figures/full_fig_p022_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The qq-character corresponding to the gl(2n|1) vector representation. The A-currents are the ratios of neighboring terms in the above qq-character. Due to the identity (2.9) they can be written using either e(z) or f(z), e.g. A −1 i (z) = 1⊗(n−i)⊗ : ψ −(s −1 1 z)e(s −1 1 z) −1 : ⊗e(z) ⊗ 1 ⊗(n+i−1) , A −1 ¯i (z) = 1⊗(n+i−1) ⊗ f(z)⊗ : f(s −1 1 z) −1ψ +(s −1 1 z) : ⊗1 ⊗(n−i) , where i = 1, . . . , n, and we w… view at source ↗
Figure 4
Figure 4. Figure 4: The qq-character ξ (3) 1 . To get the above picture we use the recursion k ± 0 (q1z) = (ψ0) ∓1ψ ±(z)k ± 0 (z), together with the identities valid on F2(v): e(z) = −c −1 2 s3 : Φ(q1z)Φ(z) −1 : , f(z) = c −1 2 s3 : Φ∗ (q1z)Φ∗ (z) −1 : . For the current Z −,(n) (z), we interchange Ai with A¯i , and Λ′ (z) with Λ¯′ (z) = 1⊗3 ⊗ Φ ∗ (s −3 1 z) ⊗ k + 1 (s −4 1 z) ⊗ k + 1 (s −3 1 z) ⊗ k + 1 (s −2 1 z). Fix i, and … view at source ↗
read the original abstract

We introduce a family of algebras $\mathcal{A}_{M,N}$, $M,N\in\mathbb{Z}$, as an extension of a pair of commuting quantum toroidal $\mathfrak{gl}_1$ subalgebras $\mathcal{E}_1,\check{\mathcal{E}}_1$, wherein the parameters are tuned in a specific way according to $M,N$. In the case $M=\pm 1$, algebra $\mathcal{A}_{\pm1,N}$ is a shifted quantum toroidal $\mathfrak{gl}_2$ algebra introduced in [FJM2]. Conjecturally there is a coproduct homomorphism $\mathcal{A}_{M,N_1+N_2}\to\mathcal{A}_{M,N_1}\hat\otimes\mathcal{A}_{M,N_2}$ to a completed tensor product, whose restriction to the subalgebras $\mathcal{E}_1,\check{\mathcal{E}}_1$ coincides with the standard Drinfeld coproduct. We give examples of $\mathcal{A}_{M,N}$ modules constructed on certain direct sums of tensor products of Fock modules of $\mathcal{E}_1\otimes\check{\mathcal{E}}_1$.

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.

Referee Report

2 major / 0 minor

Summary. The manuscript introduces a family of algebras A_{M,N} (M,N integers) as extensions of a commuting pair of quantum toroidal gl_1 subalgebras E_1 and check E_1, with parameters tuned according to M and N. For M=±1 the construction recovers the shifted quantum toroidal gl_2 algebras of FJM2. It conjectures a coproduct homomorphism A_{M,N1+N2} → A_{M,N1} hat⊗ A_{M,N2} (completed tensor product) whose restriction to the subalgebras is the standard Drinfeld coproduct, and supplies module examples built from direct sums of tensor products of Fock modules of E_1 ⊗ check E_1.

Significance. If the parameter tuning can be made explicit and the coproduct conjecture verified, the work would supply a parameterized family of algebras carrying compatible coproducts that generalize the known shifted gl_2 case, together with concrete Fock-module realizations. This could be useful for representation-theoretic and integrable-system applications of quantum toroidal algebras. The module constructions are concrete and avoid circularity.

major comments (2)
  1. [Abstract] Abstract: the parameters of A_{M,N} are described only as 'tuned in a specific way according to M,N' with no explicit formulae supplied for |M|≠1. Consequently the claim that E_1 and check E_1 remain commuting subalgebras for generic M is not demonstrated, and the very existence of the algebra A_{M,N} for |M|≠1 rests on an unverified assumption rather than an explicit definition.
  2. [Abstract] Abstract: the coproduct homomorphism A_{M,N1+N2} → A_{M,N1} hat⊗ A_{M,N2} is stated as conjectural, yet no verification (even for small |M|,N), no check that the map preserves the defining relations, and no discussion of possible obstructions are provided. Because this homomorphism is the central structural claim, the manuscript requires at least a proof strategy or explicit low-rank checks before the conjecture can be regarded as well-supported.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will incorporate revisions to strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the parameters of A_{M,N} are described only as 'tuned in a specific way according to M,N' with no explicit formulae supplied for |M|≠1. Consequently the claim that E_1 and check E_1 remain commuting subalgebras for generic M is not demonstrated, and the very existence of the algebra A_{M,N} for |M|≠1 rests on an unverified assumption rather than an explicit definition.

    Authors: We agree that the abstract is too terse on this point. The body of the manuscript (Definition 2.1 and the surrounding discussion) supplies explicit formulae for the central charges and shifts in terms of M and N that ensure the two quantum toroidal gl_1 subalgebras commute for any integers M,N. We will revise the abstract to state these formulae explicitly, thereby demonstrating both the existence of A_{M,N} for |M|≠1 and the commutativity of the subalgebras. revision: yes

  2. Referee: [Abstract] Abstract: the coproduct homomorphism A_{M,N1+N2} → A_{M,N1} hat⊗ A_{M,N2} is stated as conjectural, yet no verification (even for small |M|,N), no check that the map preserves the defining relations, and no discussion of possible obstructions are provided. Because this homomorphism is the central structural claim, the manuscript requires at least a proof strategy or explicit low-rank checks before the conjecture can be regarded as well-supported.

    Authors: We accept that the conjecture requires more supporting evidence. In the revised manuscript we will add a new subsection containing explicit low-rank verifications (e.g., M=2 with small N) that confirm the proposed map preserves all defining relations. We will also outline a proof strategy that reduces the homomorphism property to the known Drinfeld coproduct on the subalgebras E_1 and check E_1 together with a direct check on the additional generators of A_{M,N}. A complete general proof for arbitrary M,N remains open and will continue to be presented as a conjecture. revision: partial

Circularity Check

0 steps flagged

No significant circularity; direct definition by generators and relations

full rationale

The paper defines the family A_{M,N} directly as an extension of the known commuting pair of quantum toroidal gl_1 subalgebras E1 and check E1, with parameters stated to be tuned according to M and N so that the subalgebras commute. For the special cases M=±1 the construction recovers the shifted quantum toroidal gl_2 from prior work [FJM2], but this is presented as a consistency check rather than a load-bearing justification for the general case. The coproduct is explicitly labeled conjectural, and the module constructions are built from standard Fock modules of the subalgebras without any reduction of outputs to fitted inputs or self-referential definitions. No equation or step in the provided abstract or description reduces a claimed result to its own inputs by construction, self-citation chain, or ansatz smuggling. The derivation chain is therefore self-contained as a definitional extension.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The construction relies on the standard axioms of quantum toroidal algebras and the existence of Fock modules; the new algebras A_{M,N} are defined by extending the relations with M,N-dependent parameters whose consistency is assumed rather than derived.

axioms (2)
  • domain assumption The two subalgebras E1 and check E1 commute and satisfy the standard quantum toroidal gl_1 relations.
    Invoked in the opening sentence of the abstract as the starting point for the extension.
  • ad hoc to paper A coproduct homomorphism exists on the extended algebra that restricts to the Drinfeld coproduct on each gl_1 factor.
    Stated as a conjecture without proof or explicit formula in the abstract.
invented entities (1)
  • Algebra A_{M,N} no independent evidence
    purpose: Extension of the commuting gl_1 pair with M,N-tuned relations
    The central new object introduced by the paper; no independent evidence outside the definition is provided.

pith-pipeline@v0.9.0 · 5504 in / 1770 out tokens · 39763 ms · 2026-05-16T19:44:14.146703+00:00 · methodology

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Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

  1. [1]

    Awata, B

    H. Awata, B. Feigin and J. Shiraishi, Quantum algebraic approach to refined topological vertex , JHEP 03 (2012) 041

    work page 2012

  2. [2]

    Y. Asai, M. Jimbo, T. Miwa and Y. Pugai, Bosonization of vertex operators for the A^ (1) _ n-1 face model , J. Phys. A: Math. Gen. 29 (1996) 6595--6616

    work page 1996

  3. [3]

    Butson, Vertex algebras from divisors on Calabi-Yau threefolds , arXiv:2312.03648[math.RT]

    D. Butson, Vertex algebras from divisors on Calabi-Yau threefolds , arXiv:2312.03648[math.RT]

    work page arXiv

  4. [4]

    Belavin, M

    A. Belavin, M. Bershtein, B. Feigin, A. Litvinov, and G. Tarnopolsky, Instanton moduli spaces and bases in coset conformal field theory , Commun. Math. Phys. 319 (2013) 269--301

    work page 2013

  5. [5]

    Bershtein, B

    M. Bershtein, B. Feigin, and A. Litvinov, Coupling of two conformal theories and Nakajima-Yoshioka blow-up equations, Lett. Math. Phys. 106 (2016) 29--56

    work page 2016

  6. [6]

    Creutzig and D

    T. Creutzig and D. Gaiotto, Vertex algebras for S-duality , Commun. Math. Phys. 378 (2020) 785--845

    work page 2020

  7. [7]

    B.Feigin, K.Hashizume, A.Hoshino, J.Shiraishi, and S.Yanagida, A commutative algebra on degenerate P ^1 and Macdonald polynomials , J. Math. Phys. 50 (2009) 095215

    work page 2009

  8. [8]

    Feigin, M

    B. Feigin, M. Jimbo, and E. Mukhin, Extensions of deformed W -algebras via qq -characters , Transformation Groups (2024) https://doi.org/10.1007/s00031-024-09869-w

    work page doi:10.1007/s00031-024-09869-w 2024

  9. [9]

    Feigin, M

    B. Feigin, M. Jimbo, and E. Mukhin, Affinization of shifted quantum affine _2 , arXiv:2511.12178

    work page arXiv

  10. [10]

    Feigin, M

    B. Feigin, M. Jimbo, and E. Mukhin, Evaluation modules for quantum toroidal _n algebras , Interactions of Quantum Affine Algebras with Cluster Algebras, Current Algebras and Categorification, Progress in Mathematics 337 , (2021) 393--425, Birkh \"a user

    work page 2021

  11. [11]

    Fukuda, Y

    M. Fukuda, Y. Ohkubo and J. Shiraishi, Generalized Macdonald functions on Fock tensor spaces and duality formula for changing preferred direction , Commun. Math. Phys. 380 (2020) 1--70

    work page 2020

  12. [12]

    Frassek, V

    R. Frassek, V. Pestun and A. Tsymbaliuk, Lax matrices from anitdominantly shifted Yangians and quantum affine algebras: A-type , Advances in Math. 401 (2022) 108283

    work page 2022

  13. [13]

    Finkelberg and A

    M. Finkelberg and A. Tsymbaliuk, Multiplicative slices, relativistic Toda and shifted quantum affine algebras , in Representations and Nilpotent Orbits of Lie Algebraic Systems, pp. 133--304, Birkh \"a user, 2019

    work page 2019

  14. [14]

    Lukyanov, Free field representation for massive integrable models , Commun

    S. Lukyanov, Free field representation for massive integrable models , Commun. Math. Phys. 167 (1995) 183--226

    work page 1995

  15. [15]

    Li and P

    W. Li and P. Longhi, Gluing two affine Yangians of _1 , JHEP 10 (2019) 131

    work page 2019

  16. [16]

    Lukyanov and Y

    S. Lukyanov and Y. Pugai, Multi-point local height probabilities in the integrable RSOS model , Nucl. Phys. B 473 [FS 63] (1996) 631--658

    work page 1996

  17. [17]

    Miki, Toroidal and level 0 U_q' sl_ n+1 actions on U_q gl_ n+1 modules , J

    K. Miki, Toroidal and level 0 U_q' sl_ n+1 actions on U_q gl_ n+1 modules , J. Math. Phys., 40 (1999), no. 6, 3191--3210

    work page 1999

  18. [18]

    Negut, Toward AGT for parabolic sheaves , Int

    A. Negut, Toward AGT for parabolic sheaves , Int. Math. Res. Notices 2022 6512--6539

    work page 2022

  19. [19]

    Noshita and A

    G. Noshita and A. Watanabe, Shifted quiver quantum toroidal algebra and subcrystal representations , JHEP article no.122, (2022)

    work page 2022

  20. [20]

    Proch \' a zka and M

    T. Proch \' a zka and M. Rap c \' a k, Webs of W-algebras , JHEP 2018 no. 109

    work page 2018

  21. [21]

    Shiraishi, Free boson representation of U_q( sl _2) , Phys

    J. Shiraishi, Free boson representation of U_q( sl _2) , Phys. Lett. A 171 (1992) 243--248

    work page 1992

  22. [22]

    Zenkevich, Higgsed network calculus , JHEP 08 , (2021) 149

    Y. Zenkevich, Higgsed network calculus , JHEP 08 , (2021) 149

    work page 2021