arxiv: 2605.04602 · v1 · submitted 2026-05-06 · 🧮 math.RA

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On Lie Algebras with Only Inner Derivations

Bakhrom Omirov, Jie Ruan

Pith reviewed 2026-05-08 16:22 UTC · model grok-4.3

classification 🧮 math.RA

keywords Lie algebrasinner derivationsadjoint cohomologycomplete Lie algebrasnon-perfect Lie algebrassemidirect productscohomology vanishing

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The pith

New families of semidirect product Lie algebras have only inner derivations, including complete non-perfect examples.

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

The paper builds new families of non-semisimple Lie algebras as semidirect products of a semisimple algebra with a nilpotent one. It proves that the first adjoint cohomology vanishes on these families, which forces every derivation to be inner. This vanishing produces a family of complete Lie algebras whose derived algebra is a proper subspace, establishing that complete examples need not be perfect. The same approach yields perfect Lie algebras that carry a nontrivial center yet still have only inner derivations, and these examples sit in dimension 31. A factorization theorem is then applied to show that the second adjoint cohomology groups are nonzero on the constructed algebras.

Core claim

The paper shows that certain non-semisimple Lie algebras of the form semisimple semidirect nilpotent have vanishing first adjoint cohomology, so that all their derivations are inner. Using this property, a family of complete non-perfect Lie algebras is obtained. The same vanishing also produces perfect Lie algebras with nontrivial center and only inner derivations whose smallest known dimension is now 31. Analysis of the second adjoint cohomology via a factorization theorem reveals that these groups are nonvanishing.

What carries the argument

The semidirect product Lie algebra L = S ⋉ N whose first adjoint cohomology vanishes, forcing the derivation algebra to coincide with the inner derivations.

Load-bearing premise

That the chosen actions in the semidirect products produce no outer derivations beyond those already accounted for by the inner ones.

What would settle it

An explicit basis computation for the full derivation space of one concrete low-dimensional algebra in the new family, showing its dimension strictly larger than the dimension of the inner derivation space.

read the original abstract

This paper is devoted to the study of non-semisimple Lie algebras of the form $\mathcal{L} = \mathcal{S} \ltimes \mathcal{N}$ whose derivations are all inner. By generalizing the methods of Sato and Angelopoulos, we introduce new families of Lie algebras and establish the vanishing of their first adjoint cohomology. As an application, we construct a family of complete non-perfect Lie algebras, thereby providing examples that yield a positive answer to Carles' question on the existence of such algebras. In addition, we reduce the dimension of known examples of perfect Lie algebras with non-trivial center and only inner derivations to $31$. Furthermore, we employ the Hochschild--Serre factorization theorem to analyze the second adjoint cohomology groups, providing insights non-vanishing of the second adjoint cohomology groups for the algebras obtained through the paper.

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

The paper builds explicit new families of Lie algebras L = S ⋉ N with only inner derivations, answers Carles' existence question for complete non-perfect examples, and drops the dimension of known perfect ones with nontrivial center to 31.

read the letter

The main contribution is the construction of new families via a generalization of Sato-Angelopoulos techniques, together with a direct positive answer to Carles' question and a concrete reduction in dimension for the perfect case. The authors also apply the Hochschild-Serre factorization to compute or bound the second adjoint cohomology, which adds a modest but useful extra layer of information about the algebras they produce. These are concrete objects rather than abstract existence arguments, and the dimension drop to 31 is a clear improvement over prior examples if the constructions hold up. The paper is aimed squarely at people working on derivations and adjoint cohomology of finite-dimensional Lie algebras over characteristic zero. A reader already familiar with the Sato-Angelopoulos vanishing results and the structure of semidirect products will find the families and the dimension claim useful to check against their own lists of examples. The central technical step is the claim that H^1(L, L) = 0 for these new L. The stress-test note correctly flags the risk that mixed cocycles coming from the action S → Der(N) could produce outer derivations not captured by separate vanishing on S and N. If the proofs contain explicit cocycle calculations or a spectral-sequence argument that rules out those terms, the conclusion follows; otherwise the vanishing rests on an unverified extension of the earlier method. The abstract alone does not show the details, so the referee would need to see the full verification. Overall the work is solid enough on its own terms to merit sending out for review, even if the proofs require tightening on the mixed-derivation issue.

Referee Report

2 major / 2 minor

Summary. The paper studies non-semisimple Lie algebras L = S ⋉ N (S semisimple, N nilpotent) whose derivations are all inner. By generalizing Sato-Angelopoulos techniques, it establishes vanishing of the first adjoint cohomology H¹(L,L) for new families, constructs complete non-perfect examples that positively answer Carles' question on existence, reduces the dimension of known perfect examples with nontrivial center and only inner derivations to 31, and analyzes non-vanishing of H²(L,L) via the Hochschild-Serre factorization theorem.

Significance. If the vanishing results hold, the explicit families supply the first known complete non-perfect Lie algebras and lower-dimensional perfect examples with only inner derivations, directly resolving an existence question posed by Carles and improving prior dimension bounds. The Hochschild-Serre analysis of H² provides additional structural information on these algebras.

major comments (2)

[construction of families and vanishing proof] The central claim that Der(L) = ad(L) for the new families L = S ⋉ N rests on generalizing the Sato-Angelopoulos cohomology vanishing to the semidirect-product setting. The manuscript must supply an explicit verification (via Hochschild-Serre spectral sequence or direct cocycle computation) that no mixed derivations arise from the action homomorphism S → Der(N); without this, the conclusion that all derivations are inner does not follow from the separate vanishing on S and N.
[perfect Lie algebras with nontrivial center] The dimension-31 perfect example with nontrivial center is obtained by a specific choice of semidirect product; the paper should include a direct check (e.g., explicit basis and bracket table or computer-assisted computation) that H¹(L,L) = 0 and Z(L) ≠ 0 for this algebra, confirming the reduction from prior examples.

minor comments (2)

[preliminaries] Notation for the action map and the resulting bracket in S ⋉ N should be introduced once and used consistently; currently the semidirect product is defined in multiple places with slightly varying symbols.
[Hochschild-Serre analysis] The statement that the second cohomology is non-vanishing should be accompanied by at least one concrete cocycle or dimension computation for a representative algebra in the family.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will make the requested additions to strengthen the proofs and verifications.

read point-by-point responses

Referee: The central claim that Der(L) = ad(L) for the new families L = S ⋉ N rests on generalizing the Sato-Angelopoulos cohomology vanishing to the semidirect-product setting. The manuscript must supply an explicit verification (via Hochschild-Serre spectral sequence or direct cocycle computation) that no mixed derivations arise from the action homomorphism S → Der(N); without this, the conclusion that all derivations are inner does not follow from the separate vanishing on S and N.

Authors: We agree that an explicit verification is required to rigorously exclude mixed derivations arising from the action. In the revised manuscript, we will add a dedicated subsection applying the Hochschild-Serre spectral sequence to the extension 0 → N → L → S → 0. This will show that the E₂-page terms corresponding to mixed cocycles vanish under the given hypotheses on the representation S → Der(N), thereby confirming that H¹(L, L) = 0 follows from the separate vanishings on S and N together with the compatibility of the semidirect product action. revision: yes
Referee: The dimension-31 perfect example with nontrivial center is obtained by a specific choice of semidirect product; the paper should include a direct check (e.g., explicit basis and bracket table or computer-assisted computation) that H¹(L,L) = 0 and Z(L) ≠ 0 for this algebra, confirming the reduction from prior examples.

Authors: We thank the referee for this suggestion. In the revised version we will include an explicit basis for the 31-dimensional algebra, the full set of nonzero Lie brackets, and a direct (hand or computer-assisted) computation verifying that every derivation is inner and that the center is one-dimensional and nontrivial. This will provide the requested confirmation that the example is indeed perfect with nontrivial center and only inner derivations. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit constructions and generalized cohomology proofs are independent of inputs

full rationale

The paper defines explicit families of semidirect products L = S ⋉ N, then generalizes Sato-Angelopoulos methods to compute H^1(L,L)=0 directly via Hochschild-Serre factorization and cocycle analysis. This establishes Der(L)=ad(L) without any parameter fitting, self-definition of the target property, or reduction of the vanishing statement to a prior result by the same authors. The positive answer to Carles' question and the dimension reduction to 31 follow as consequences of these new examples rather than tautological re-labeling. No load-bearing step equates a derived claim to its own construction or a self-citation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard axioms of Lie algebras (bilinearity, skew-symmetry, Jacobi identity), the definition of semidirect product, the definition of inner derivation via the adjoint representation, and the definition of adjoint cohomology groups. No free parameters or invented entities are indicated in the abstract; the new families are constructed explicitly rather than postulated.

axioms (2)

standard math Lie algebra axioms: bilinear, skew-symmetric bracket satisfying Jacobi identity
Invoked throughout the study of derivations and cohomology.
domain assumption Semidirect product structure L = S ⋉ N with S semisimple and N nilpotent
Central to the form of algebras considered and to the application of Hochschild-Serre factorization.

pith-pipeline@v0.9.0 · 5437 in / 1508 out tokens · 45757 ms · 2026-05-08T16:22:28.182218+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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