{"id":"b87c4779-b2f7-452e-b8b7-23d20868b409","arxiv_id":"2607.06512","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The fully nilpotent and seminilpotent cohomological Hall algebras of A² are commutative, identified with symmetric algebras on sQ[s,u], and their equivariant deformations are enveloping algebras of Rees Lie algebras from W_1+∞^+.","lead":"This paper proves that two nilpotent versions of the cohomological Hall algebra of the plane A² — previously known only partially — are actually commutative. A reader might care because these algebras encode counts of matrix pairs and link to W-algebra symmetries, so understanding when they become commutative clarifies a key contrast with the non-nilpotent case.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Key perverse-filtration bound in Theorems 1.1/1.2 is not fully justified: the Lie bracket on BPS⊗H* is defined via a non-canonical projection that need not respect the perverse filtration.","rationale":"The reader's weakest_assumption already flags perverse-filtration compatibility after applying ı!_nil and ı!_SN, and my concern is a precise way that compatibility can fail: the sheaf-level Lie bracket is not literally the CoHA product but a product followed by a projection onto BPS⊗H* parallel to an arbitrary complement E. The paper notes that after derived global sections the bracket is independent of E, but the proof uses the filtration before taking sections. This is a real soft spot in the argument as written, though not a demonstrated falsehood: the theorem itself is plausible and for the fully nilpotent case is independently supported by [Mel+23]. A careful choice of E via semisimplicity may repair the proof. The reader's verdict of CONDITIONAL therefore remains appropriate; I do not move to ACCEPT or REJECT. I chose 'partial' because the reader identified filtration compatibility but not the specific projection issue.","tokens_in":15153,"tokens_out":15394,"duration_ms":150804,"concrete_test":"Verify the filtration claim by choosing E compatibly with the perverse filtration via the semisimple decomposition of AA2 from [Dav25, Theorem A], then recompute the sheafified bracket component BPS[-2m] ⊡ BPS[-2n] → BPS[-2k] in the C*-equivariant Hom of Lemma 4.1(4.1) for small m,n (e.g. m=n=0 and m=n=1), and apply ı!_nil to the resulting morphism. If any component with k=m+n+1 survives ı!_nil and preserves cohomological degree, the proof's bound is false; if all such components vanish, the perverse-filtration claim is corroborated and the proof can be repaired.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorems 1.1 and 1.2 in Section 4 rests on the claim that the Lie bracket on BPS_{A2} ⊗ H*_{C*}(pt) respects the perverse filtration, and hence maps BPS[-2m] ⊡ BPS[-2n] to ⊕_{k≤m+n} BPS[-2k] (fully nilpotent) or ⊕_{k<m+n} BPS[-2k] (seminilpotent). However, in Section 2 this sheaf-level Lie bracket is defined only as the actual CoHA product followed by the projection onto the summand BPS⊗H* parallel to an arbitrary complement E of AA2. The product respects the perverse filtration, but the projection need not: an arbitrary direct-sum complement E need not be compatible with the perverse filtration, so the composite can increase perverse degree. The text acknowledges that the sheaf-level bracket may depend on E and that only after derived global sections is it independent of E. But the filtration bound is applied before taking derived global sections and before applying ı!_nil/ı!_SN. If the projection increases perverse degree by one, the target bound becomes k≤m+n+1, and the cohomological-degree counting that forces the bracket to vanish no longer works: for the fully nilpotent case the source and target degrees would then match. Thus the central vanishing argument has a genuine gap as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the fully nilpotent and seminilpotent cohomological Hall algebras (CoHAs) of A^2. The main claims are that both algebras are commutative, with primitive Lie algebra isomorphic to s Q[s,u] (Theorems 1.1 and 1.2), and that under a C^*-action with weights (1,-1) the equivariant versions are enveloping algebras of Rees Lie algebras associated with filtrations on W^+_{1+∞} (Theorem 1.3). The proofs use the sheafified CoHA and BPS Lie algebra machinery, combine perverse-filtration and cohomological-degree constraints to show the Lie bracket vanishes, and use explicit C^*-equivariant Hom computations for the equivariant deformation.","tokens_in":15470,"tokens_out":21781,"duration_ms":184592,"significance":"If the results hold, they give a clean contrast with Davison's noncommutative description of the non-nilpotent CoHA of A^2 and provide the first complete descriptions of the nilpotent CoHAs in this basic example. The equivariant Rees-Lie-algebra description is elegant and likely to be useful. The paper also contains a useful Milnor–Moore statement over PIDs. However, the central proof as written has a gap concerning the compatibility of the sheaf-level projection with the perverse filtration, and there are technical issues in the equivariant component computation. These are repairable, but they affect load-bearing steps.","major_comments":[{"comment":"The central vanishing argument uses the assertion that the sheaf-level Lie bracket on BPS_{A^2}⊗H^*_{C^*}(pt) respects the perverse filtration. But §2 defines this bracket as the CoHA product followed by projection onto BPS_{A^2}⊗H^*_{C^*}(pt) parallel to an arbitrary complement E, and explicitly notes that this projection may depend on E. A projection parallel to an arbitrary complement need not be filtration-preserving, so the claimed bound BPS[-2m]⊡BPS[-2n] → ⊕_{k≤m+n} BPS[-2k] (and the strict version k<m+n in Theorem 1.2) is not established by the argument that the product respects the perverse filtration. Since the subsequent cohomological degree count in both proofs relies exactly on this bound, the proof has a gap. A fix would require choosing E compatible with the perverse filtration (possible via the semisimple decomposition of AA^2) and proving the projected bracket is filtered","section":"§4, proof of Theorem 1.1 and Theorem 1.2"},{"comment":"Lemma 4.3 states that the restriction maps (4.4) and (4.5) are multiplication by -ℏ and -ℏ^2 for all m,n,k. This is false as stated: for k>m+n-1 in (4.4) and for k>m+n+1? More precisely, for k>m+n-1 the source Hom space in Lemma 4.1 vanishes, while the displayed target Hom spaces can be nonzero. A linear map from the zero space to a nonzero space cannot be 'multiplication by -ℏ' or '-ℏ^2'. The lemma is only meaningful on the range where the source Hom is nonzero. The proof of Theorem 1.3 appears to use only the range k≤m+n-1, so the computation may survive, but the statement and its application need to be restricted to the nonzero range.","section":"§4, Lemma 4.3"},{"comment":"The claimed isomorphism Φnil appears to have an off-by-one error with respect to the definition of RF^nil[W^+_{1+∞}]. For r=0, Φnil(p^nil_{d,a}) = -z^d D^{a+1} t^{a+1}. By the definition of RF^nil as Span{z^d D^A t^i | d≥1, A≥0, i≥A+1}, this element has A=a+1 and i=a+1, so i=A, violating i≥A+1. Thus the image is not contained in RF^nil as defined. The analogous map ΦSN is consistent, which suggests a typo in the shift, but as written the isomorphism statement for the fully nilpotent equivariant Lie algebra is not correct.","section":"§4, proof of Theorem 1.3"}],"minor_comments":[{"comment":"'As mentionned' should be 'As mentioned'. Also, the paragraph after Theorem 1.1 notes that the fully nilpotent commutativity can be deduced from [Mel+23]; this is relevant context and might be stated earlier.","section":"§1.2.1"},{"comment":"The symbol C(gl_d) is used for both the commuting stack and its affinization/GIT quotient. This is potentially confusing in the Cartesian diagrams of Section 3; consider a notational distinction.","section":"§2"},{"comment":"The use of a and b for the shifts in the Hom computations collides notationally with the exponents in W^+_{1+∞}. It is harmless but may hinder readability; renaming the shifts would help.","section":"§4, Lemma 4.1"}],"recommendation":"major_revision","confidential_remarks":"The fully nilpotent result is acknowledged by the author as following from [Mel+23]; the main novelty appears to be the seminilpotent case and the equivariant Rees descriptions. The gaps identified in Section 4 are local and likely fixable, but they affect the central proof, so a major revision is appropriate. The paper also relies heavily on several external and partly overlapping works by the same group; the editor may wish to ensure those references are stable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper proves that the fully nilpotent and seminilpotent CoHAs of A2 are commutative, with primitive Lie algebra sQ[s,u], and gives a C*-equivariant description in terms of Rees Lie algebras of W+1+∞. These are good results, and Theorem 1.2 especially appears new. The writing is clear, and the author is honest about dependencies, even noting that Theorem 1.1 can be derived from Mellit–Minets–Schiffmann–Vasserot. Lemma 4.3's Euler class computations also look correct.\n\nThe problem is in the proof of Theorems 1.1 and 1.2. The argument needs the sheafified Lie bracket on BPS⊗H*C*(pt) to send BPS[-2m] ⊡ BPS[-2n] into ⊕_{k≤m+n} BPS[-2k] (or k<m+n for seminilpotent). The author says this follows because the CoHA product respects the perverse filtration and \"the same becomes immediately true for the Lie bracket.\" But the sheaf-level Lie bracket is not just the product; it is the product followed by projection onto the BPS summand parallel to an arbitrary complement E. The paper explicitly notes that this projection may depend on E, and only becomes canonical after derived global sections. Nothing guarantees that an arbitrary E is compatible with the perverse filtration. If the projection increases perverse degree by one, the target bound becomes k≤m+n+1, and the cohomological-degree counting no longer forces vanishing. So the proof as written has a real gap.\n\nThis is very likely fixable: semisimplicity of the underlying complex [Dav25, Theorem A] should allow one to choose a complement E that is a direct sum of shifted perverse sheaves, making the projection filtered. But the author needs to say that and justify it. As is, the vanishing argument does not go through.\n\nThe paper deserves a serious referee. The results are important and probably correct, but the non-equivariant commutativity proofs need to be tightened. Theorem 1.1 has independent support from [Mel+23], so the gap is less damaging there; Theorem 1.2 carries more weight.","headline":"Commutativity of the nilpotent CoHAs of A2 is likely true, but the proof of the main vanishing claim has a genuine gap: the sheaf-level bracket is defined via an arbitrary projection that need not respect the perverse filtration.","tokens_in":15992,"tokens_out":11230,"would_cite":true,"duration_ms":98581,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16G20","14N35","14F43"],"pacs":[],"model":"deepseek-v4-flash","headline":"The fully nilpotent and seminilpotent cohomological Hall algebras of A^2 are commutative, with primitive Lie algebra sQ[s,u], despite the non-commutativity of the ordinary A^2 CoHA.","keywords":["cohomological Hall algebra","nilpotent CoHA","BPS Lie algebra","perverse filtration","commuting variety","A^2","Jordan quiver","W_{1+∞}"],"falsifier":"Compute the Lie bracket of the two primitive generators s and s·u in the fully nilpotent CoHA; the theorem predicts it vanishes in the Borel–Moore homology of the stack of nilpotent commuting 2×2 matrices (C_nil(gl_2)). A direct convolution-product calculation finding a nonzero class there would refute commutativity.","tokens_in":15011,"feed_emoji":"🧮","tokens_out":10867,"duration_ms":85191,"temperature":0.7,"pith_summary":"The paper proves that both nilpotent variants of the cohomological Hall algebra (CoHA) of the affine plane A^2 are commutative, in sharp contrast to the ordinary CoHA of A^2, which is non-commutative and related to the Lie algebra W_{1+∞} of differential operators on C*. The fully nilpotent CoHA (built from pairs of commuting nilpotent matrices) and the seminilpotent CoHA (one matrix nilpotent) both have primitive Lie algebra isomorphic to the abelian Lie algebra sQ[s,u], making each CoHA a symmetric (polynomial) algebra on that space. The proof shows the Lie bracket must vanish after nilpotent restriction: the perverse filtration bounds the perverse degree of the bracket, while the cohomological grading fixes its total degree, and the two constraints become incompatible on the nilpotent loci. A C*-equivariant version identifies the equivariant nilpotent CoHAs as enveloping algebras of Rees Lie algebras obtained from filtrations on W^+_{1+∞}.","feed_headline":"Forcing nilpotency makes the A^2 Hall algebra commutative","feed_subtitle":"Both nilpotent variants become polynomial algebras, unlike the non-commutative non-nilpotent CoHA.","key_machinery":"The proof's load-bearing tool is a double constraint on the Lie bracket of the affinized BPS Lie algebra: it is filtered with respect to the perverse filtration on the BPS cohomology, and simultaneously graded with respect to the cohomological degree. On the nilpotent loci these two constraints are brought into conflict: the perverse-filtration bound allows the bracket to land only in cohomological degrees strictly lower than the source's degree, forcing it to be zero. Supporting cast: the nilpotent restriction functors ı!_nil and ı!_SN, the cohomological integrality isomorphisms, and the graded Milnor–Moore theorem, which turns the abelian primitive Lie algebra into a symmetric algebra.","core_discovery":"The central claim is that the affinized BPS Lie algebras of A^2, when restricted to the fully nilpotent or seminilpotent locus, are abelian: they are both isomorphic to the abelian Lie algebra sQ[s,u], with the cohomological degrees of the generators scaled differently in the two cases. Consequently the fully nilpotent CoHA and the seminilpotent CoHA are commutative algebras, each isomorphic to Sym(sQ[s,u]). The proof shows that the Lie bracket on the affinized BPS Lie algebra respects the perverse filtration — after restriction, the bracket of a generator of cohomological degree 2+2m with one of degree 2+2n can land only in perverse degrees at most m+n — while the bracket is also graded wit","pith_inferences":["The degree-mismatch mechanism suggests a template: any situation where nilpotent restriction raises the cohomological degree of BPS generators while the perverse-filtration bound stays the same will yield a commutative CoHA. A natural test is other one-vertex quivers, such as cyclic quivers, whose BPS Lie algebras are similarly concentrated.","Theorem 1.3 locates the nilpotent CoHAs inside the deformation (Rees) picture of W^+_{1+∞}; this hints that the full 2-torus equivariant nilpotent CoHAs, which lack a coproduct, might still be governed by the same Rees Lie algebras but with a nonstandard module structure.","The polynomial structure of the nilpotent CoHAs could make them attractive for explicit enumerative computations, where the complexity of the non-nilpotent CoHA has been a bottleneck; the commutative presentation gives immediate access to characters and Hilbert series."],"forward_implications":["The fully nilpotent CoHA of A^2 is a polynomial algebra on generators s^k u^l, with dimension and cohomological degrees (k, 2+2l).","The seminilpotent CoHA is the same polynomial algebra, but the cohomological degree of s^k is 0, so the degree of s^k u^l is 2l; the different grading does not change the algebra.","The C*-equivariant nilpotent CoHAs are exactly the enveloping algebras of the Rees Lie algebras of W^+_{1+∞} for the nilpotent and seminilpotent filtrations, matching the known description of the non-nilpotent equivariant CoHA.","The commutative structure gives explicit generators-and-relations descriptions of the nilpotent CoHAs of A^2, in contrast to the non-nilpotent case where the description involves the non-commutative degenerate W_{1+∞} algebra."],"fun_headline_variants":["Nilpotent A^2 Hall algebras turn commutative","A^2 nilpotent CoHAs: polynomial, unlike non-nilpotent","Perverse filtration kills noncommutativity in A^2 CoHAs","When nilpotent, A^2 CoHAs become abelian"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The argument assumes that applying the nilpotent restriction does not break the perverse-filtration upper bound on the Lie bracket; if nilpotent restriction allowed the bracket to reach higher perverse degrees, the cohomological-degree mismatch would disappear and commutativity would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Nilpotent A^2 Hall algebras turn commutative","A^2 nilpotent CoHAs: polynomial, unlike non-nilpotent","Perverse filtration kills noncommutativity in A^2 CoHAs","When nilpotent, A^2 CoHAs become abelian"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000313,"raw_usage":{"total_tokens":1645,"prompt_tokens":801,"completion_tokens":844,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":545,"completion_tokens_details":{"reasoning_tokens":763}},"tokens_in":545,"tokens_out":844,"duration_ms":7046,"temperature":1.0,"reasoning_tokens":763,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T04:23:28.244449+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the Lie bracket of the two primitive generators s and s·u in the fully nilpotent CoHA; the theorem predicts it vanishes in the Borel–Moore homology of the stack of nilpotent commuting 2×2 matrices (C_nil(gl_2)). A direct convolution-product calculation finding a nonzero class there would refute commutativity.","supporting_citations":[],"review_version":2}