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Commutativity of nilpotent cohomological Hall algebras of $\mathbf{A}^2$

T0 review · 3 major / 3 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2607.06512 v2 pith:UDFQYQ7X submitted 2026-07-07 math.RT math.AG

Lucien Hennecart This is my paper · ORCID
classification math.RTmath.AG MSC 16G2014N3514F43
keywords cohomologicalHallalgebranilpotentCoHABPSLieperversefiltrationcommutingvarietyA^2JordanquiverW_{1+∞}
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

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+∞}.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

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.

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 (3)
  1. [§4, proof of Theorem 1.1 and Theorem 1.2] 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
  2. [§4, Lemma 4.3] 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.
  3. [§4, proof of Theorem 1.3] 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.
minor comments (3)
  1. [§1.2.1] '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.
  2. [§2] 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.
  3. [§4, Lemma 4.1] 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.

Circularity Check

0 steps flagged · score 2.0 of 10

No definitional circularity: the nilpotent commutativity theorems are derived from external BPS Lie algebra and integrality inputs, not assumed. The filtration/projection step in Thm 1.1 is a proof gap, not a circular reduction.

full rationale

The derivation chain is not circular. Theorem 1.1's conclusion (ĝnil ≅ sQ[s,u], vanishing bracket) is obtained from the cohomological integrality isomorphism and the BPS sheaf computation g_nil ≅ ⊕ Q{0_d}[-2], not assumed: the vector-space identification fixes the generators, and the bracket vanishes by a degree/filtration count. Theorem 1.2 uses the same integrality plus the less-perverse-filtration input. The Milnor–Moore steps are standard and do not smuggle commutativity into the hypotheses. Theorem 1.3 is an identification with Rees Lie algebras via Davison's [Dav22, Thm C] and the Gysin restrictions of Lemma 4.3; the bracket coefficients are transferred from W+1+∞, not postulated to vanish. Self-citations appear: [HJ26, Thm 1.1] is invoked for the vanishing of the associated graded bracket in Thm 1.2, and [Dav+26] for the factorization coproduct. These are substantive structural results about the non-nilpotent/less perverse picture and do not have nilpotent commutativity as their statement, so they are dependence rather than circularity. I flag a genuine proof gap, though not a circular one: Section 2 defines the sheaf-level bracket as product followed by projection onto BPS⊗H* parallel to an arbitrary complement E and admits the projection may depend on E; the proof of Thm 1.1 then asserts this composite respects the perverse filtration without justifying that the chosen projection is filtered. If the projection raised perverse degree, the degree count would fail. This is a missing argument, not an input-output identity. Also, the paper notes Theorem 1.1 can be deduced from [Mel+23], an external source, which supports that the statement itself is not an artifact of a self-referential construction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new particles or abstract entities. However, it depends on a network of established constructions (BPS sheaves, filtrations, Rees algebras) whose properties are imported from cited works. The main new content is the structural identification of the nilpotent CoHAs.

assumptions (5)
  • domain assumption Cocommutative bialgebra structure and Milnor–Moore theorem apply to nilpotent CoHAs.
    Used in Section 1.2.1 and Section 3 to identify CoHAs with enveloping algebras of primitive Lie algebras. The paper cites [Dav+26, §6.2], [Jin24], [HJ26] and Appendix A, but does not prove the existence of factorization coproducts for the nilpotent variants in full detail.
  • domain assumption Perverse filtration respects the Lie bracket on BPS ⊗ H_*^{C*}(pt).
    Used in the proof of Theorem 1.1 and Theorem 1.2 to bound the perverse degree of the bracket. The paper states this follows from the CoHA product being a morphism of constructible complexes, but relies on standard t-structure properties.
  • domain assumption BPS Lie algebra of A² is commutative and its affinized version has prescribed graded pieces.
    Used to control the Lie bracket when applying ı!_nil and ı!_SN. The paper cites [Dav25, Proposition 7.3], [Dav23, Theorem D], [Dav22].
  • domain assumption C*-equivariant cohomological integrality isomorphisms hold for the nilpotent restrictions.
    Used in the proof of Theorem 1.3 to identify the equivariant CoHAs as enveloping algebras and to compute the bracket components. The paper states these follow by restriction from the non-nilpotent integrality, but the detailed proof is omitted.
  • domain assumption The C*-equivariant sheafified Lie bracket is determined by a finite number of scalar components.
    Corollary 4.2 asserts that the sheafified bracket is determined by the induced bracket on global sections. The proof uses Lemma 4.1 and integrality, but the identification of the bracket components as morphisms between simple summands relies on semisimplicity results [Dav25, Theorem A].

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Pith. "Pith review of Commutativity of nilpotent cohomological Hall algebras of $\mathbf{A}^2$." pith.science (2026). https://pith.science/paper/UDFQYQ7X

@misc{pith2026260706512,
  author       = {Pith},
  title        = {Pith review of: Commutativity of nilpotent cohomological Hall algebras of $\mathbfA^2$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UDFQYQ7X}},
  note         = {Machine review of arXiv:2607.06512}
}
abstract

In this paper, we prove that both the seminilpotent and the fully nilpotent CoHAs of $\mathbf{A}^2$ are commutative. This result is in strong contrast with the CoHA of $\mathbf{A}^2$ without nilpotency conditions, previously studied by Davison, which is related to the Lie algebra $W_{1+\infty}$ of differential operators on $\mathbf{C}^*$. The latter is highly noncommutative. Our proof combines two constraints on the Lie bracket on the affinized BPS Lie algebra: it is filtered with respect to the perverse filtration and it is graded with respect to the cohomological degree. In the case of the Jordan quiver and nilpotent CoHAs, these constraints force the Lie bracket to vanish. We also describe the equivariant nilpotent CoHAs in the presence of the action of a one-dimensional torus rescaling the first coordinate of $\mathbf{A}^2$ with weight $1$ and the second with weight $-1$. In this case, one obtains enveloping algebras of Rees Lie algebras associated with the nilpotent and the seminilpotent filtrations on the Lie algebra $W_{1+\infty}^+$, reminiscent of the description of the equivariant non-nilpotent CoHA given by Davison.

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. The loop-nilpotent cohomological Hall algebra

    math.RT 2026-07 conditional novelty 7.0 of 10

    Loop-nilpotent CoHAs of tripled quivers are isomorphic to an explicit integral shuffle algebra, yielding generators, Coulomb-branch surjections, BPS characterizations, and a new Kac-polynomial formula.

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Works this paper leans on

4 extracted references · 2 linked inside Pith · cited by 1 Pith paper

  1. [30]

    Commuting varieties of semisimple Lie algebras and alge- braic groups

    [Ric79] R William Richardson. “Commuting varieties of semisimple Lie algebras and alge- braic groups”. In: Compositio Mathematica 38.3 (1979), pp. 311–327. [RS17] Jie Ren and Yan Soibelman. “Cohomological Hall algebras, semicanonical bases and Donaldson–Thomas invariants for 2-dimensional Calabi–Yau categories (with an appendix by Ben Davison)”. In: Algeb...

    1979

  2. [1074]

    On two cohomological Hall algebras

    [YZ20] Yaping Yang and Gufang Zhao. “On two cohomological Hall algebras”. In: Pro- ceedings of the Royal Society of Edinburgh Section A: Mathematics 150.3 (2020), pp. 1581–1607. Laboratoire Ami´enois de Math´ematique Fondamentale et Appliqu´ee, CNRS UMR 7352, Universit´e de Picardie Jules Verne, 33 rue Saint Leu, 80000 Amiens, France Email address : lucie...

    2020

  3. [1975]

    On the number of points of nilpotent quiver varieties over finite fields

    REFERENCES 19 [BSV20] Tristan Bozec, Olivier Schiffmann, and Eric Vasserot. “On the number of points of nilpotent quiver varieties over finite fields”. In: Annales scientifiques de l’ ´Ecole normale sup´ erieure53.6 (2020), pp. 1501–1544. [Dav+26] Ben Davison, Lucien Hennecart, Tasuki Kinjo, Olivier Schiffmann, and Eric Vasserot. “Hecke operators on sympl...

    arXiv 2020

  4. [2002]

    Coher- ent sheaves on surfaces, COHAs and deformed W1+∞-algebras

    [Mel+23] Anton Mellit, Alexandre Minets, Olivier Schiffmann, and Eric Vasserot. “Coher- ent sheaves on surfaces, COHAs and deformed W1+∞-algebras”. In: arXiv preprint arXiv:2311.13415 (2023). [Min20] Alexandre Minets. “Cohomological Hall algebras for Higgs torsion sheaves, moduli of triples and sheaves on surfaces”. In: Selecta Mathematica 26.2 (2020), p

    arXiv 2023

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