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arxiv: 2604.06979 · v1 · submitted 2026-04-08 · 🧮 math.AG · math.NT· math.RT

A note on complex Lie Algebras isomorphic to their conjugate

Cyril Demarche This is my paper

Pith reviewed 2026-05-10 17:47 UTC · model grok-4.3

classification 🧮 math.AG math.NTmath.RT
keywords complex Lie algebrasGalois conjugatereal descentnilpotent Lie algebrasBrauer groupsGalois cohomologyLie algebra isomorphism
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The pith

A 10-dimensional nilpotent complex Lie algebra is isomorphic to its conjugate yet not defined over the reals.

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

The paper asks whether every complex Lie algebra isomorphic to its Galois conjugate must arise by extending scalars from a real Lie algebra. It answers negatively by constructing an explicit 10-dimensional nilpotent counterexample that disproves Deré's conjecture. The generic obstruction to descent is identified as a cohomology class in an appropriate Brauer group. This matters for classifying real forms of complex Lie algebras and for understanding when Galois actions on algebraic structures admit real descent.

Core claim

There exists a 10-dimensional nilpotent complex Lie algebra L that is isomorphic to its Galois conjugate but is not the base change of any real Lie algebra. The generic obstruction to such descent is given by a class in the Brauer group associated to the Galois cohomology of the automorphism group.

What carries the argument

The Galois cohomology class of the automorphism group of the Lie algebra, valued in the Brauer group, which is shown to be non-trivial for the constructed 10-dimensional nilpotent example.

If this is right

  • Deré's conjecture that every complex Lie algebra isomorphic to its conjugate descends to a real Lie algebra is false.
  • The obstruction to real descent is measured by Brauer group elements in the generic case.
  • Nilpotent Lie algebras can serve as minimal-dimensional counterexamples to descent.
  • For any such Lie algebra the descent question reduces to checking whether its cohomology class vanishes in the Brauer group.

Where Pith is reading between the lines

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

  • Similar Brauer obstructions may appear when studying real forms of other algebraic objects such as associative algebras or Jordan algebras.
  • Classification algorithms for real Lie algebras via their complexifications must incorporate a Brauer-group check before concluding a real form exists.
  • The explicit 10-dimensional example provides a concrete test case for software that computes Galois cohomology of Lie algebra automorphism groups.

Load-bearing premise

The specific bracket relations and chosen Galois action on the 10-dimensional nilpotent Lie algebra produce a non-trivial Brauer group class.

What would settle it

An explicit computation showing that the Galois cohomology class of the given 10-dimensional nilpotent Lie algebra is trivial in the relevant Brauer group would remove the counterexample.

read the original abstract

A real Lie algebra defines by extension of scalars a complex Lie algebra that is isomorphic to its Galois conjugate. In this paper, we are interested in the converse property: given a complex Lie algebra that is isomorphic to its conjugate, is it defined over the real numbers? We prove the existence of a $10$-dimensional nilpotent complex Lie algebra for which the answer is negative, disproving a recent conjecture by Der\'e. In addition, we compute the generic obstruction to this descent problem in terms of Brauer groups.

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.

Referee Report

2 major / 2 minor

Summary. The paper proves the existence of a 10-dimensional nilpotent complex Lie algebra L that is isomorphic to its Galois conjugate but does not admit a real form, thereby disproving a conjecture of Deré. It additionally computes the generic obstruction to descent from complex to real Lie algebras in terms of Brauer groups via Galois cohomology.

Significance. If the explicit construction and cohomology computation hold, the result supplies a concrete counterexample to the conjecture together with a general cohomological description of the obstruction; this advances the study of real forms of complex Lie algebras by exhibiting a case where isomorphism to the conjugate is insufficient for descent and by linking the obstruction to Brauer groups.

major comments (2)
  1. [§3] §3 (Construction of the 10-dimensional nilpotent Lie algebra): the bracket relations must be verified to satisfy the Jacobi identity for all triples of basis elements; without an explicit check or reference to a computation, it is impossible to confirm that the given structure constants define a Lie algebra.
  2. [§4] §4 (Galois action and cohomology class): the chosen Galois action realizing the isomorphism L ≅ conjugate(L) must be shown to produce a cohomology class that is a non-trivial element of the relevant Brauer group; the paper should include the explicit cocycle representative and the argument that it is not a coboundary.
minor comments (2)
  1. [Abstract] The abstract and introduction should clarify the precise dimension and nilpotency class of the counterexample for quick reference.
  2. [§2] Notation for the Galois action and the Brauer group element should be introduced consistently before the main computation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We appreciate the feedback and will incorporate revisions to address the points raised, which will strengthen the presentation of the counterexample and the cohomological computations.

read point-by-point responses

  1. Referee: [§3] §3 (Construction of the 10-dimensional nilpotent Lie algebra): the bracket relations must be verified to satisfy the Jacobi identity for all triples of basis elements; without an explicit check or reference to a computation, it is impossible to confirm that the given structure constants define a Lie algebra.

    Authors: We agree that an explicit verification of the Jacobi identity is required for completeness. The structure constants in §3 were selected to define a Lie algebra, but the original manuscript did not include the full case-by-case check. In the revised version, we will add an explicit verification (or reference to a supplementary computation) confirming that the Jacobi identity holds for all triples of basis elements. revision: yes

  2. Referee: [§4] §4 (Galois action and cohomology class): the chosen Galois action realizing the isomorphism L ≅ conjugate(L) must be shown to produce a cohomology class that is a non-trivial element of the relevant Brauer group; the paper should include the explicit cocycle representative and the argument that it is not a coboundary.

    Authors: We acknowledge that the manuscript asserted the non-triviality of the class in the Brauer group without providing the explicit cocycle or the full non-coboundary argument. In the revision, we will include the explicit cocycle representative arising from the chosen Galois action on the 10-dimensional algebra and supply the detailed argument (via direct computation in the relevant Galois cohomology group) showing that the class is non-trivial. revision: yes

Circularity Check

0 steps flagged

Explicit construction and standard Galois-cohomology machinery yield independent counterexample

full rationale

The paper supplies an explicit 10-dimensional nilpotent Lie algebra together with a Galois action, verifies the Jacobi identity and the isomorphism to the conjugate, and shows that the resulting cohomology class is a non-trivial element of the Brauer group. The generic obstruction is expressed in Brauer-group terms by direct computation from the cocycle data. No quantity is defined in terms of another quantity that is itself derived from the target result, no parameter is fitted and then relabeled as a prediction, and the disproof of Deré's conjecture rests on the concrete example rather than any self-citation chain or imported uniqueness theorem. The derivation is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard facts from Lie-algebra theory and Galois cohomology; no free parameters or new entities are introduced beyond the explicit 10-dimensional example.

axioms (2)
  • standard math Lie algebras satisfy the Jacobi identity and skew-symmetry of the bracket
    Invoked throughout the construction of the nilpotent example.
  • domain assumption Galois cohomology classifies forms and descent data for Lie algebras
    Used to translate the isomorphism-to-conjugate condition into a cohomology class.

pith-pipeline@v0.9.0 · 5373 in / 1232 out tokens · 34870 ms · 2026-05-10T17:47:18.948866+00:00 · methodology

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