{"id":"48a892a7-a2df-4b6d-9731-70ec2809e2c2","arxiv_id":"2512.21750","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A family of algebras A_{M,N} is introduced as extensions of commuting quantum toroidal gl_1 pairs, with module examples and a conjectured coproduct that restricts to the Drinfeld coproduct.","lead":"The paper defines a new family of algebras A_{M,N} that extend a pair of commuting quantum toroidal gl_1 subalgebras with parameters tuned by integers M and N. For M equal to plus or minus one these recover known shifted gl_2 algebras, and the authors construct some modules while conjecturing a coproduct map.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Parameter tuning for A_{M,N} relations lacks explicit verification for general M,N beyond |M|=1 cases","rationale":"The reader's weakest_assumption correctly isolates the unverified tuning step. Because the paper presents the coproduct only as a conjecture and gives module constructions that presuppose the algebra exists, the absence of an explicit consistency check for general M,N is the single load-bearing gap. This does not invalidate the known |M|=1 cases or the module examples, but it keeps the general claim at the level of a well-motivated conjecture rather than a fully established construction.","tokens_in":1798,"tokens_out":370,"duration_ms":18736,"concrete_test":"For M=2, N=1, substitute the claimed parameter tuning into the defining relations of A_{2,1} and recompute the commutators between all generators of E1 and check E1; verify whether they all vanish identically. If any non-zero term survives, the algebra is not well-defined for that (M,N).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central construction defines A_{M,N} by extending the pair E1, check E1 with M,N-dependent parameters chosen so the subalgebras commute and the conjectural coproduct homomorphism exists. The abstract states that parameters are 'tuned in a specific way according to M,N' and notes that A_{±1,N} recovers the known shifted quantum toroidal gl_2, but supplies no explicit formulae for the parameters when |M|≠1 and no direct check that the cross-commutators vanish or that the coproduct preserves all relations. Without these, the very existence of the algebra A_{M,N} for generic M remains an assumption rather than a demonstrated fact.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","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.","tokens_in":1991,"tokens_out":548,"duration_ms":31532,"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":[{"comment":"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.","section":"Abstract"},{"comment":"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.","section":"Abstract"}],"minor_comments":[],"recommendation":"major_revision","confidential_remarks":"The manuscript is at an early stage; the main result is a conjecture whose feasibility is not yet demonstrated beyond the |M|=1 case already in the literature. The work may be more suitable for a short note once the parameter formulae and a verification of the coproduct on generators are supplied."},"author_rebuttal":{"model":"grok-4.3","summary":"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.","responses":[{"response":"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_made":"yes","referee_comment":"[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."},{"response":"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_made":"partial","referee_comment":"[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."}],"tokens_in":1474,"tokens_out":511,"duration_ms":19934,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"This paper introduces a family of algebras A_{M,N} for integers M and N as extensions of a pair of commuting quantum toroidal gl_1 subalgebras, with parameters tuned according to M and N. In the special case where M is plus or minus one, it recovers the shifted quantum toroidal gl_2 algebra from your prior work. They also construct examples of modules on direct sums of tensor products of Fock modules for the subalgebras. Conjecturally, there is a coproduct map from A_{M, N1+N2} to the completed tensor product of A_{M,N1} and A_{M,N2} that restricts to the standard Drinfeld coproduct on the subalgebras. What stands out as new is the general A_{M,N} family for arbitrary M and N, along with these module examples. The definitions appear direct, extending the known gl_1 pair without obvious loops back to fitted data. The module constructions on Fock modules seem original and provide concrete examples in this setting. The paper does a good job of setting up the algebras explicitly and giving module examples that build from standard objects. This could be helpful for anyone looking at representations of these extended algebras. The soft spots are around the parameter tuning and the coproduct. The abstract mentions tuning parameters in a specific way but does not give the explicit expressions for general M and N, only noting the recovery for M equals plus or minus one. This makes it difficult to confirm that the subalgebras commute for other values or that the relations are consistent. The coproduct is presented as conjectural without any verification or counterexample. The stress-test note correctly points out that the existence of A_{M,N} for generic M relies on this tuning working out, which is not shown. If the full text has the details and checks, that would address the main concern. This work is aimed at specialists in quantum groups, toroidal algebras, and their representation theory. A reader already familiar with Drinfeld coproducts and Fock modules in this context would find the new family and module examples worth examining. I recommend sending it for peer review. The constructions are specific enough that referees can evaluate the parameter choices and test the conjectures, which could lead to a solid contribution once the gaps are filled.","headline":"The paper introduces a new family of algebras A_{M,N} extending commuting quantum toroidal gl_1 pairs with conjectural coproduct and Fock module examples, but leaves parameter tuning unverified for general M,N.","tokens_in":2500,"tokens_out":555,"would_cite":false,"duration_ms":26708,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[],"headline":"Quantum toroidal gl_1 extensions and coproduct conjectures unrelated to RS cost or forcing chain","alignment":"orthogonal","rationale":"The paper's core machinery (parameter-tuned A_{M,N} extensions of commuting E_1, check E_1 subalgebras, fused currents, screened intertwiners on Fock modules, conjectural Drinfeld-style coproduct) operates entirely within the domain of quantum affine/toroidal algebras and vertex operator constructions. No structures parallel J(x) = ½(x + x^{-1}) − 1, φ-ladder, 8-tick periodicity, or parameter-free constant derivations. The tuning of parameters for commutativity and coproduct homomorphism is an algebraic ansatz with no connection to recognition-cost functional equations or absolute-floor forcing.","tokens_in":75644,"confidence":"high","tokens_out":183,"duration_ms":7975,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":["17B37","17B65"],"pacs":[],"model":"grok-4.3","headline":"Algebras A_{M,N} extend commuting quantum toroidal gl_1 pairs by tuning parameters to M and N with a conjectured coproduct.","keywords":["quantum toroidal algebra","gl_1","commuting subalgebras","coproduct","Fock modules","shifted quantum toroidal gl_2","algebra extensions","Drinfeld coproduct"],"falsifier":"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.","tokens_in":2681,"feed_emoji":"","tokens_out":773,"duration_ms":34393,"temperature":0.7,"pith_summary":"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.","feed_headline":"Algebras extend commuting quantum toroidal gl_1 pairs with tuned parameters","feed_subtitle":"M and N tune the relations to keep the gl_1 subalgebras commuting and support a conjectured coproduct matching the Drinfeld version on each.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":[],"fun_headline_variants":["Commuting quantum toroidal gl1 pairs extended by A_MN","Tuned A_MN extends commuting quantum toroidal gl1 pairs","A_MN extends commuting quantum toroidal gl1 with M N tuning","Parameter tuning yields A_MN extensions of gl1 pairs"],"cache_read_input_tokens":64,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Commuting quantum toroidal gl1 pairs extended by A_MN","Tuned A_MN extends commuting quantum toroidal gl1 pairs","A_MN extends commuting quantum toroidal gl1 with M N tuning","Parameter tuning yields A_MN extensions of gl1 pairs"]},"model":"grok-4.3","cost_usd":0.01067,"raw_usage":{"total_tokens":4666,"prompt_tokens":742,"num_sources_used":0,"completion_tokens":70,"cost_in_usd_ticks":106703000,"prompt_tokens_details":{"text_tokens":742,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3854,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":742,"tokens_out":70,"duration_ms":29783,"temperature":1.0,"reasoning_tokens":3854,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-16T19:44:14.146703+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"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.","supporting_citations":[],"review_version":1}