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REVIEW 4 major objections 4 minor 29 references

On realisations of the Steenrod algebras

T0 review · 4 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read The Steenrod algebra cannot be realized as the universal enveloping algebra of any Lie superalgebra, so the definition of enveloping algebra in positive characteristic needs revision.

desk verdict The p>2 computation in Theorem 3.1 cancels to zero under the Adem relations, so the headline non-realization claim fails as written, but the Deligne letters and open problems are worth preserving. read the letter →

arxiv 2509.09443 v2 pith:7FWHRE32 submitted 2025-09-11 math.AG

Alexei Lebedev , Dimitry Leites This is my paper · ORCID
classification math.AG MSC 55S10
keywords SteenrodalgebrauniversalenvelopingLiesuperalgebrapositivecharacteristicPBWtheoremdividedpowersformalgroupschemeAdemrelations
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 claims that, for every prime p, the Steenrod algebra A(p) is not isomorphic to the universal enveloping algebra U(g) of any Z-graded Lie superalgebra g, under the conventional definitions (usual or restricted), provided the identity operation P^0 (or Sq^0) is a scalar. The argument uses the PBW theorem to reconstruct the dimensions and parities of the hypothetical g from the graded dimensions of A(p), then shows that the element P^1 (or Sq^1) would have to lie in g with properties that force a contradiction: in characteristic 2, squaring behavior mismatches, and for odd p, a nested commutator would have to occupy a weight for which g has no room. The paper concludes that the standard notion of enveloping algebra is inadequate in positive characteristic and advocates generalized enveloping algebras U(g;N) built from divided-power generators. An appended commentary suggests instead that the Steenrod algebra is best understood as the hyperalgebra of a formal group scheme, not as an enveloping algebra.

What carries the argument

The load-bearing device is the PBW theorem used as a dimension counter: U(g) has a basis of ordered monomials in the elements of a basis of g, so the dimension of each graded component U(g)_k equals the dimension of g_k plus the dimension of the (super)symmetric polynomials built from the lower graded pieces G_{k-1}. The paper uses this to compute g_k recursively from the known graded dimensions of A(p)_k, and then confronts the resulting g with the Steenrod algebra's own relations — the squaring of odd elements in characteristic 2 and the p-th power and bracket behavior of P^1 in odd characteristic — to produce a contradiction. A second mechanism is the paper's proposed family of enveloping

What would settle it

Evaluate the double commutator [P^1,[P^1,P^p]] in A(p) for an odd prime using the standard Adem relations with the paper's conventions; the paper claims it equals 2λP^{p+1}P^1, a nonzero element, and a direct expansion that yields zero would invalidate the proof's key step for p>2.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is a no-go theorem: no Z-graded Lie superalgebra g admits a grading-preserving isomorphism from the Steenrod algebra A(p) to U(g), whether for the usual or the restricted universal enveloping algebra, when P^0 (or Sq^0) is a scalar. The proof combines a PBW-based induction that uniquely determines the graded dimensions and parities of g_k from the known dimensions of A(p)_k with explicit bracket computations inside A(p). For p=2, the generators Sq^1 and Sq^2 would have to produce squaring products that contradict the Adem relations; for p>2, P^1 must lie in g, its p-th power restricts the algebra to a restricted enveloping algebra, and then the

Load-bearing premise

The p>2 half of the proof relies on the claim that the nested commutator [P^1,[P^1,P^p]] is a nonzero element of weight 2(p-1)(p+2); if the Adem relations force this commutator to zero, the contradiction for odd primes does not follow.

Editorial extensions

If this is right

  • Differential-operator realizations of A(p) discussed in the paper cannot be promoted to isomorphisms with enveloping algebras; they remain embeddings, so the 'realization' problem needs a different target.
  • The proposed family U(g;N) of divided-power enveloping algebras is a concrete path to repairing PBW theory in positive characteristic, connecting the Steenrod question to representation theory of vectorial Lie algebras.
  • The PBW dimension-reconstruction argument applies to any graded algebra with known dimensions, giving a necessary condition for a graded algebra to be an enveloping algebra; this is a reusable test.
  • The group-scheme alternative suggests a functorial characterization of A(p) as the algebra of natural operations on mod-p cohomology, with the dual being the affine algebra of the automorphism group of the formal additive group.

Reading between the lines

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

  • A natural testable extension is whether some member of the proposed family U(g;N), with a suitable shearing vector, does realize A(p); the paper leaves this open, and a positive example would both validate the new definition and supply the realization the authors originally sought.
  • The proof's p>2 step depends on a specific Adem-relation computation of a nested commutator; if that computation fails, the theorem might still hold by a different argument, but the present proof would need repair — possibly a contradiction using restricted p-th powers alone.
  • The group-scheme viewpoint suggests that other formal groups give rise to 'Steenrod-like' algebras via their hyperalgebras; studying them could yield new families of cohomology operations, a direction the appended commentary gestures toward but does not develop.
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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

4 major / 4 minor

Summary. The paper argues that the mod-p Steenrod algebra A(p) cannot be realized as the universal enveloping algebra U(g) of any Z-graded Lie superalgebra under conventional definitions. It reviews notions of enveloping algebras in positive characteristic, proposes new divided-power variants, proves a formal non-realization theorem (Theorem 3.1) for grading-preserving isomorphisms, and reproduces Deligne's correspondence. The central proof for p>2 hinges on a nested commutator computation in A(p).

Significance. If the main theorem were correct, it would answer a long-standing folklore question and motivate the paper's proposed redefinition of U(g). The paper also has useful historical and conceptual content: the Deligne letters, the dimension-counting argument in Theorem 3.1, and the discussion of divided-power enveloping algebras. However, the proof of the main theorem for odd primes contains a concrete false computation: the nested commutator claimed to be nonzero actually vanishes under the standard Adem relations. The theorem is therefore unsupported, and the abstract's blanket no-realisation claim exceeds what is proved.

major comments (4)
  1. [§3, proof of Theorem 3.1 (p>2)] The decisive computation is incorrect. The proof claims [P^1,[P^1,P^p]] = 2(λ^2P^{p+2}−λP^{p+1}P^1+λP^pP^2)=2λP^{p+1}P^1≠0. Applying the standard Adem relations used elsewhere in the same proof, with P^0=λ, gives P^pP^2 = −λP^{p+2}+P^{p+1}P^1 for odd p. Substituting into the displayed expression yields 2(λ^2P^{p+2}−λP^{p+1}P^1+λ(−λP^{p+2}+P^{p+1}P^1))=0. Thus there is no nonzero element of weight 2(p−1)(p+2), the contradiction collapses, and Theorem 3.1 is not proved for odd p. The equality '2(...)=2λP^{p+1}P^1' is not an identity in A(p).
  2. [Eq. (4) and its use in §3] The displayed Adem relation for P^aP^b in (4) sums over i=1..[a/p], omitting the i=0 term. The subsequent proof relies on i=0 terms: e.g., P^1P^p=λP^{p+1}, P^2P^p=λP^{p+2}, and P^pP^2=−λP^{p+2}+P^{p+1}P^1. If (4) is taken literally, these reductions are not justified; if (4) is corrected to include i=0, the commutator computation in the preceding comment cancels. The proof is internally inconsistent with the stated presentation of A(p).
  3. [Abstract, §1, and Theorem 3.1] The abstract and Section 1 state that no Lie superalgebra realizes A(p) as U(g). Theorem 3.1 only excludes grading-preserving isomorphisms f:A(p)→U(g) under two extra hypotheses: P^0 (or Sq^0) is a scalar, and the parity of g_k equals the parity of k. The theorem does not address non-grading-preserving isomorphisms, nor the alternative enveloping definitions discussed in Section 2 and Deligne's Section 4. The gap between the advertised claim and the proven statement should be acknowledged and the conclusion restricted accordingly.
  4. [§3, p=2 case] The p=2 argument is not fully rigorous. The proof refers to a table for small degrees and to 'similar computations' for restricted algebras, but does not derive the key claims, e.g. that (L_2)^2=Sq^3Sq^1 can hold only in a restricted algebra or that it must be an element of g. Given the nonstandard definition of Lie superalgebras in characteristic 2 in §1.3, the relationship between the squaring operation and the product in U(g) needs to be spelled out. The p=2 part of the theorem is therefore not established even independently of the odd-primary gap.
minor comments (4)
  1. [Eq. (4)] In the second displayed relation, the variable in the binomial coefficient appears as 'a-pu' and should be 'a-pi'. The bracket notation '[a/p]' should be defined as the floor function.
  2. [§3, table] The table would benefit from a caption explaining the entries: L_i denotes a chosen basis element of g_i, and products such as L_2^2 or L_2^2L_1 are elements of U(g); without this, the reader cannot check the PBW dimension counts.
  3. [§4] Deligne's letters are interesting, but they are reproduced as a verbatim letter and are not clearly marked as an appendix. The authors should state explicitly which conclusions in Section 4, if any, are used in the body of the paper.
  4. [§2] The proposed U(g;N) algebras are never defined. Since the paper argues that the conventional enveloping algebra is inadequate, a precise definition of the proposed replacement (even for a model example) would make the discussion more usable.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: Theorem 3.1 is a direct reductio computation against the standard Steenrod algebra presentation and PBW, with no fitted parameters or load-bearing self-citations.

full rationale

The paper's central claim (Theorem 3.1) is proved by assuming an isomorphism f:A(p)→U(g), then using PBW to determine the possible dimensions of the graded pieces of g, and finally deriving a contradiction from explicit bracket computations using the Adem relations (Eqs. 4–5). This is a standard reductio argument: the target result is not an input, no parameter is fitted to a subset of data and then renamed a prediction, and the argument is checked directly against the defining relations of the Steenrod algebra. Self-citations such as [BLLS] and [BLLS1] appear only as background for definitions and classifications in characteristic 2, not as evidence for Theorem 3.1. Deligne's appended comments are explicitly comments, hints, and open problems; they are not used to establish the theorem. Even if the p>2 nested-commutator computation contains a mathematical error (as suggested by the supplied note), that is a correctness concern, not circularity: the problematic value is computed from the Adem relations, not assumed. No circular step of any enumerated kind is present.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The theorem's proof depends on PBW-type dimension counts and on computations inside A(p). No free parameters are fitted and no new physical or algebraic entities are postulated; the proposed U(g;N) algebras are an open problem, not asserted objects.

assumptions (4)
  • domain assumption PBW theorem and filtration dimension formula hold for common and restricted enveloping algebras of Lie superalgebras in characteristic p, with dimensions of (super)symmetric polynomials governing U(g)_k.
    Used in Section 3.1 to determine dim g_k by induction from dim A(p)_k. This is standard for Lie algebras but delicate for Lie superalgebras at p=2, and the paper does not fully justify it.
  • standard math The Steenrod algebra A(p) is presented by the Adem relations as in Section 1.2, including the standard i=0 terms.
    The paper's printed Eq. (4) omits the i=0 term, yet the proof later appears to use it. A correct version of the Adem relations is needed for the commutator computations.
  • standard math The graded dimensions of A(p) are known and are used to read off the dimensions of g_k.
    Invoked in Section 3.1 with only a partial table for p=2 and a terse assertion for p>2.
  • domain assumption The theorem assumes g is Z-graded with parity(g_k)=k mod 2 and that the isomorphism is grading preserving.
    This is a hypothesis of Theorem 3.1, but it is much narrower than the abstract's unqualified claim about all realizations.

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Cite this review

Pith. "Pith review of On realisations of the Steenrod algebras." pith.science (2026). https://pith.science/paper/7FWHRE32

@misc{pith2026250909443,
  author       = {Pith},
  title        = {Pith review of: On realisations of the Steenrod algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7FWHRE32}},
  note         = {Machine review of arXiv:2509.09443}
}
read the original abstract

The Steenrod algebra can not be realised as an enveloping of any Lie superalgebra. We list several problems that suggest a need to modify the definition of the enveloping algebra, for example, to get rid of certain strange deformations which we qualify as an artefact of the inadequate definition of the enveloping algebra in positive characteristic. P. Deligne appended our paper with his comments, hints and open problems.

Discussion (0). Continue with ORCID to comment.

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