Commutative Lie algebras and commutative cohomology in characteristic $2$

Pasha Zusmanovich; Viktor Lopatkin

arxiv: 1907.03690 · v1 · pith:5M53I7QZnew · submitted 2019-07-08 · 🧮 math.KT · math.RA

Commutative Lie algebras and commutative cohomology in characteristic 2

Viktor Lopatkin , Pasha Zusmanovich This is my paper

Pith reviewed 2026-05-25 00:43 UTC · model grok-4.3

classification 🧮 math.KT math.RA

keywords Lie algebrascharacteristic 2Chevalley-Eilenberg cohomologysymmetric cochainscommutative cohomologycohomology of Lie algebras

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The pith

In characteristic 2, a symmetric cochain complex defines a version of Chevalley-Eilenberg cohomology for Lie algebras.

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

The paper constructs and discusses a cohomology theory for Lie algebras over fields of characteristic 2 by replacing the usual alternating cochains with symmetric ones inside the Chevalley-Eilenberg framework. In characteristic 2 the sign -1 coincides with 1, so the classical alternating condition loses its distinguishing power and a symmetric replacement becomes natural. A reader would care because the resulting theory may still detect extensions, derivations, and deformations of Lie algebras even when the standard alternating complex collapses or becomes trivial. If the construction succeeds, it supplies concrete cochain complexes and cohomology groups that can be computed directly for commutative Lie algebras in this setting.

Core claim

The authors define a commutative cohomology theory by taking the symmetric algebra on the dual of a Lie algebra L instead of the exterior algebra, equipping the resulting cochain space with a differential that mimics the classical Chevalley-Eilenberg formula but respects the symmetric product; they then study the resulting cohomology groups for commutative Lie algebras over fields of characteristic 2.

What carries the argument

The symmetric cochain complex obtained by replacing the exterior product with the symmetric product in the Chevalley-Eilenberg construction.

If this is right

The new complex yields well-defined cohomology groups that can be calculated for any commutative Lie algebra in characteristic 2.
These groups are expected to classify commutative extensions and deformations in the same formal way the classical groups classify ordinary ones.
The construction remains functorial with respect to Lie algebra homomorphisms.
It provides a setting in which the usual long exact sequences and spectral sequences of Lie algebra cohomology may be re-derived.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

One could test whether the symmetric cohomology agrees with ordinary cohomology after base change to characteristic zero for algebras that lift.
The same replacement might be tried in other cohomology theories that rely on alternation, such as group cohomology in characteristic 2.
If the theory detects new invariants, it could be applied to the classification of simple commutative Lie algebras over finite fields of characteristic 2.

Load-bearing premise

That the resulting symmetric cochain complex produces a cohomology theory whose algebraic properties remain sufficiently close to those of the classical alternating theory to make the replacement useful.

What would settle it

An explicit computation for a low-dimensional commutative Lie algebra over a field of characteristic 2 in which the symmetric cohomology groups fail to classify central extensions that are known to exist by other means.

read the original abstract

We discuss a version of the Chevalley--Eilenberg cohomology in characteristic $2$, where the alternating cochains are replaced by symmetric ones.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Short technical note swapping to symmetric cochains for Chevalley-Eilenberg cohomology in char 2; the key check that d²=0 holds is not visible in the abstract.

read the letter

This paper proposes a version of Chevalley-Eilenberg cohomology in characteristic 2 that replaces alternating cochains with symmetric ones, and the title indicates it also treats commutative Lie algebras in that setting. The central move is to adapt the standard cochain complex so that the signs work when -1 equals 1. If the authors supply an explicit differential and show that its square vanishes, using symmetry together with the appropriate form of the Jacobi identity, then the construction gives a usable cohomology theory for this narrow case. That replacement itself looks like the main new element relative to the literature they cite. The work is straightforward and stays within homological algebra without claiming larger reorganizations. The soft spot is precisely the one flagged in the stress test. The classical proof that d²=0 leans on alternation to produce canceling minus signs; symmetry removes those, so it is not obvious that the same cancellation occurs or that a modified formula for d restores it. The abstract states the idea but shows neither the differential nor any verification or example. If the full paper contains a clear algebraic check or a small computation confirming d²=0, the concern disappears; otherwise the claim that a cohomology theory exists rests on an unproven identity. This is a niche note aimed at people already working on Lie algebra cohomology or homological algebra in positive characteristic. A reader looking for a concrete variant in char 2 could find it helpful as a reference, but it will not interest anyone outside that circle. I would send it to peer review so that experts can verify the differential and the square-zero property in a few pages. The topic is legitimate and the scope is modest, so referee time is appropriate if the algebra checks out.

Referee Report

1 major / 0 minor

Summary. The manuscript proposes and discusses a version of Chevalley-Eilenberg cohomology for Lie algebras over fields of characteristic 2, obtained by replacing the usual alternating cochains with symmetric cochains; it also introduces the notion of commutative Lie algebras in this setting.

Significance. If the construction yields a valid cohomology theory with useful properties, it would supply a homological tool adapted to characteristic 2, where the classical alternation map vanishes identically. Such a theory could be relevant to deformation theory, algebraic K-theory, or the study of restricted Lie algebras, but its utility depends on the extent to which the resulting cohomology groups are computable and functorial.

major comments (1)

The central claim is that a cohomology theory exists once alternating cochains are replaced by symmetric ones. For this to hold, the proposed differential on the symmetric cochain spaces must satisfy d² = 0. The manuscript must supply an explicit verification of this identity (including the precise formula for d and the algebraic cancellations that occur in characteristic 2), as the classical proof relies on alternation and the Jacobi identity in a way that does not automatically transfer to the symmetric case.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need for an explicit verification of the key identity. We address the major comment below and will incorporate the requested details in a revised version of the manuscript.

read point-by-point responses

Referee: The central claim is that a cohomology theory exists once alternating cochains are replaced by symmetric ones. For this to hold, the proposed differential on the symmetric cochain spaces must satisfy d² = 0. The manuscript must supply an explicit verification of this identity (including the precise formula for d and the algebraic cancellations that occur in characteristic 2), as the classical proof relies on alternation and the Jacobi identity in a way that does not automatically transfer to the symmetric case.

Authors: We agree that the manuscript should contain an explicit, self-contained verification that the proposed differential on symmetric cochains squares to zero. In the revision we will add a dedicated subsection that states the precise coboundary formula (adapted to the symmetric setting and the commutative Lie bracket), expands d² on a general symmetric cochain, and isolates the cancellations that hold in characteristic 2. These cancellations rely on the symmetry of the cochains together with the modified Jacobi identity that defines commutative Lie algebras; the argument does not invoke alternation and therefore differs from the classical Chevalley–Eilenberg proof. We will also record the precise algebraic identities used at each step so that the verification is fully transparent. revision: yes

Circularity Check

0 steps flagged

No circularity; definitional construction of symmetric-cochain cohomology in char 2 is self-contained.

full rationale

The paper proposes a version of Chevalley-Eilenberg cohomology by replacing alternating cochains with symmetric ones in characteristic 2. This is framed as a direct construction and discussion rather than a derivation that reduces to fitted parameters, self-citations, or prior results by the same authors. No load-bearing equations, uniqueness theorems, or ansatzes are visible that collapse by construction to the inputs. The central claim is the existence and basic properties of the new complex, which stands as an independent definitional proposal. Verification of d²=0 (if present) would constitute an algebraic proof internal to the paper, not a circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities can be identified.

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