BPS Lie algebras, perverse filtrations and shuffle algebras

Andrei Negu\c{t}; Shivang Jindal

REVIEW 2 major objections 1 minor 1 cited by

The BPS Lie algebra of any quiver with zero potential has an explicit description via limit conditions on shuffle polynomials.

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2026-07-13 15:20 UTC pith:FPVM4FH2

load-bearing objection Abstract-only claim of an explicit algebraic description of BPS Lie algebras for zero-potential quivers via perverse filtrations and shuffle limits; looks like real progress in the subfield if the proofs hold. the 2 major comments →

arxiv 2604.00124 v2 pith:FPVM4FH2 submitted 2026-03-31 math.RT math.AGmath.QA

BPS Lie algebras, perverse filtrations and shuffle algebras

Shivang Jindal , Andrei Negu\c{t} This is my paper

classification math.RT math.AGmath.QA MSC 17B3714F0816G20

keywords BPS Lie algebracohomological Hall algebraperverse filtrationshuffle algebraquiver with potentialrepresentation theory

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper claims that the BPS Lie algebra attached to any quiver with zero potential can be written down explicitly. The authors relate the perverse filtration on the cohomological Hall algebra to concrete limit conditions on polynomials that appear in the shuffle presentation of that algebra. With that identification in hand, the graded pieces that define the BPS Lie algebra become algebraic rather than geometric, so one can describe them by inspecting which polynomials satisfy the limits. The same method yields a partial description of the perverse filtration for quivers with nonzero potential, and the authors conjecture that the description becomes complete for the special class of tripled quivers equipped with their canonical cubic potential. A sympathetic reader cares because the BPS Lie algebra packages enumerative and representation-theoretic information that is usually hard to access; an explicit shuffle description turns that package into something one can compute with.

Core claim

For any quiver with zero potential, the BPS Lie algebra admits an explicit algebraic description obtained by identifying the geometric perverse filtration on its cohomological Hall algebra with certain limit conditions on the polynomials that appear in the shuffle algebra presentation.

What carries the argument

The identification between the perverse filtration on the cohomological Hall algebra and limit conditions on shuffle polynomials: this dictionary converts a geometric filtration into an algebraic one and thereby produces the explicit generators of the BPS Lie algebra.

Load-bearing premise

That the geometric perverse filtration on the cohomological Hall algebra is completely captured, when the potential is zero, by the stated algebraic limit conditions on polynomials in the shuffle presentation.

What would settle it

For a concrete quiver with zero potential whose BPS Lie algebra is already known by other means, check whether the polynomials that satisfy the authors' limit conditions recover exactly those known generators and no others.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

Abstract-only claim of an explicit algebraic description of BPS Lie algebras for zero-potential quivers via perverse filtrations and shuffle limits; looks like real progress in the subfield if the proofs hold.

read the letter

The one thing you need to know is that this paper claims a clean, explicit description of the BPS Lie algebra for every quiver with zero potential, by matching the geometric perverse filtration on the COHA to certain polynomial limit conditions in the shuffle algebra. That is the main theorem. For nonzero potential they get a partial description and conjecture completeness for the usual tripled quivers with cubic potential.

What is actually new is the comparison itself: turning the geometric filtration into concrete algebraic conditions that let you write down the BPS Lie algebra without further geometry. If the identification works, it organizes a central object in geometric representation theory and DT theory and should make more computations feasible. The abstract is careful about the scope—full for zero potential, partial plus conjecture otherwise—so they are not overselling.

The soft spot is obvious and proportionate: we only have the abstract. No proofs, no examples, no explicit formulas to check. The load-bearing step is precisely that the perverse filtration is completely captured by those polynomial limits; the reader correctly flags this as the weakest assumption, and the stress-test finds nothing worse. Circularity risk looks low from the framing (two a priori different structures being compared), but without the text we cannot audit it. Everything else—novelty within the subfield, significance for people who work with COHAs and BPS algebras—looks solid on the face of it.

This is for specialists in geometric representation theory, quantum groups, and Donaldson–Thomas theory who already care about COHAs and BPS Lie algebras. A serious referee in that circle should see it. I would send it to peer review rather than desk-reject; the claim is sharp enough and the authors flag the limits of what they prove. Whether the proofs survive is for the referees to decide.

Referee Report

2 major / 1 minor

Summary. The manuscript claims an explicit description of the BPS Lie algebra of any quiver with zero potential, obtained by identifying the geometric perverse filtration on the cohomological Hall algebra with certain algebraic limit conditions on polynomials in a shuffle presentation. It further asserts a partial description of the perverse filtration for arbitrary potential, together with a conjecture that this description is complete for tripled quivers equipped with the canonical cubic potential.

Significance. If the claimed identification holds, the work would supply a concrete algebraic model for BPS Lie algebras of zero-potential quivers, linking the geometric perverse filtration on COHA to shuffle-algebra data and thereby making these structures more accessible for computation and comparison. The partial results and the conjecture for nonzero potentials would likewise clarify the extent to which the same algebraic conditions capture the filtration in the presence of a potential. Such a description would be a useful contribution to geometric representation theory and Donaldson–Thomas theory.

major comments (2)

[Abstract] Only the abstract is available for review. The central load-bearing claim—that the geometric perverse filtration on the COHA is completely captured, for zero potential, by the stated algebraic limit conditions on polynomials—cannot be checked for correctness, range of validity, or hidden assumptions without the proofs, definitions, and examples that would appear in the body of the paper. A full technical assessment is therefore impossible on the present material.
[Abstract] The abstract asserts a partial description for arbitrary potential and a completeness conjecture for tripled quivers with canonical cubic potential, but supplies no statement of what is proved versus what is conjectured, nor any indication of the evidence supporting the conjecture. Without the corresponding sections, the strength of these claims cannot be evaluated.

minor comments (1)

[Abstract] The abstract is concise but does not name the precise class of shuffle algebras or the form of the limit conditions; a one-sentence expansion would help readers locate the result relative to existing literature on COHA and BPS algebras.

Circularity Check

0 steps flagged

No significant circularity detectable from abstract-only material; claimed route compares a priori distinct structures.

full rationale

Only the abstract is available, so no equations, definitions, or self-citations can be inspected for reduction-by-construction. The abstract presents the main result as an explicit description of the BPS Lie algebra for zero-potential quivers obtained by relating the geometric perverse filtration on the COHA to algebraic limit conditions on polynomials in a shuffle presentation. That framing is the non-circular pattern: two a priori different structures are compared, and the identification is offered as a theorem rather than as a redefinition of one object by the other. There is no visible fitted parameter renamed as a prediction, no uniqueness theorem imported from the authors, and no ansatz smuggled in via citation. Residual risk that the limit conditions were reverse-engineered to match known BPS data cannot be substantiated without the body of the paper; manufacturing circularity from that possibility would violate the hard rules. Score 0 with empty steps is therefore the honest finding.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

Abstract-only audit. The paper sits in standard geometric representation theory: it relies on the existence and basic properties of cohomological Hall algebras, perverse filtrations on moduli of quiver representations, shuffle-algebra presentations, and the definition of BPS Lie algebras. No free numerical parameters are indicated. No new physical particles or forces are introduced; the “entities” are algebraic constructions already in the literature. The main ad-hoc-to-paper ingredient, invisible in detail, is the precise form of the “limit conditions on polynomials” used to match the perverse filtration.

axioms (5)

domain assumption Existence and standard properties of the cohomological Hall algebra (COHA) of a quiver (with or without potential).
The entire comparison is internal to COHA; the abstract takes COHA as given background.
domain assumption The perverse filtration on COHA is well-defined via the geometry of moduli spaces of quiver representations.
Central object being matched to algebraic limit conditions; assumed from geometric representation theory.
domain assumption Shuffle-algebra presentations of (parts of) COHA are available and involve polynomials to which limit conditions can be applied.
The explicit description is stated in terms of those presentations and limits.
domain assumption Standard definitions of BPS Lie algebras associated to quivers / DT theory.
The object being described; taken from prior literature in the field.
ad hoc to paper The specific “limit conditions on polynomials” correctly characterize the perverse filtration for zero potential.
This identification is the paper’s load-bearing technical move; its precise form is not in the abstract and is treated as the authors’ contribution rather than standard background.

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0 comments

read the original abstract

We give an explicit description of the BPS Lie algebra of any quiver with zero potential, by relating the perverse filtration on the cohomological Hall algebra with certain limit conditions on polynomials. Our results also give a partial description of the perverse filtration for arbitrary potential, which we conjecture is complete in the case of tripled quivers with canonical cubic potential.

discussion (0)

Forward citations

Cited by 1 Pith paper

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

The loop-nilpotent cohomological Hall algebra
math.RT 2026-07 conditional novelty 7.0

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.