Post-Lie nonexistence with a semisimple target

Abstract

Let g be a finite-dimensional complex unimodular Lie algebra and n a complex semisimple Lie algebra of the same dimension. We prove that if two Lie algebra homomorphisms j₁, j₂ : g → n have invertible difference, then g is semisimple. The integrated left-right action has a holomorphic infinitesimal determinant. Unimodularity makes this determinant invariant, and its constancy on an open orbit extends to the whole target group. Transitivity and simple connectedness then identify the source and target manifolds; their compact-form homology rules out a nonzero source radical. Consequently a perfect nonsemisimple complex source and a semisimple target admit no post-Lie algebra structure.

The idea in plain English

A Lie algebra describes infinitesimal symmetries. A post-Lie product is an extra operation that makes two Lie brackets on the same vector space compatible. This paper asks when that compatibility is impossible.

The result rules out every finite-dimensional complex pair whose source is perfect but not semisimple and whose target is semisimple. In this language, perfect means generated by its brackets, while semisimple means having no nonzero solvable ideal. The paper actually proves a stronger obstruction using the weaker source condition called unimodularity.

The striking step is geometric: a local compatibility condition would force two associated symmetry spaces to be the same manifold. Their topology then detects the source’s solvable part and shows that it must vanish. This turns a problem about algebraic identities into a short obstruction from geometry and homology.

The argument addresses the semisimple-target conjecture described by Burde and Dekimpe, extending earlier special cases. It adapts a determinant method with an explicit predecessor in Baues’s work and uses established Lie theory; historical first priority is not claimed.

The precise result

Theorem 1: let g and n be complex Lie algebras of the same finite dimension. If tr(ad_g x) = 0 for every x ∈ g, n is semisimple, and homomorphisms j₁, j₂ : g → n have an invertible linear difference, then g is semisimple.

Corollary 3: if a post-Lie structure exists on (g, n), with g unimodular and n semisimple, then g must be semisimple. Every perfect Lie algebra is unimodular, so a perfect nonsemisimple source admits no such structure with any semisimple target.

The homomorphisms need not be injective, algebraic, or have closed integrated images. Their invertible difference is the transversality assumption. The zero-dimensional case is included, with the zero algebra semisimple by convention.

The result supplies the semisimple-target conclusion conjectured on printed p. 2689 of Oberwolfach Report 46/2023. The separate conjectures there concerning nilpotent targets are outside this result.

Why the result holds

Integrate the two homomorphisms to simply connected complex groups. Their left-right action has a globally holomorphic determinant measuring whether its infinitesimal orbit directions span the target tangent space.

Unimodularity makes this determinant invariant. It is a nonzero constant on an open orbit, so the holomorphic identity theorem makes it that same constant everywhere. All orbits are open, forcing transitivity on the connected target.

The stabilizer is closed and discrete. The orbit map is therefore a covering, and simple connectedness makes it a diffeomorphism of the source and target manifolds.

A simply connected solvable radical is contractible. The source retracts onto the compact form of its semisimple Levi factor, while the target has a nonzero homology class in the degree of its full complex dimension. A nonzero source radical would make those homology groups incompatible.

For a semisimple post-Lie target, inner derivations yield the established pair j₁ = R + I and j₂ = R. Their difference is the identity, so the obstruction applies.

Scope and limitations

The published theorem is stated for finite-dimensional complex Lie algebras. It does not provide a separate treatment of arbitrary real forms, positive characteristic or infinite dimension.

Unimodularity cannot simply be omitted. Remark 4 supplies a three-dimensional nonunimodular solvable source with two homomorphisms into sl₂(C) whose difference is invertible.

A merely reductive target is not covered. Central directions change both the inner-derivation reduction and the topological obstruction; earlier literature contains perfect-source examples with reductive nonsemisimple targets.

The orbit map identifies manifolds, and is not asserted to be a group homomorphism. The proof’s conclusion is semisimplicity of the source; a Lie algebra isomorphism is not inferred solely from that manifold identification.

The determinant method, transverse-pair formulation and Lie-theoretic and topological inputs have substantial predecessors. The literature comparison is bounded, and the original body of Onishchik’s 1969 decomposition paper was not accessible in this review. No exhaustive novelty or first-priority certificate is given.

Exact finite diagnostic checks accompany the paper. They cannot establish the global integration, transitivity, covering-space or homology arguments, which depend on the analytical proof and its cited classical results.

Status and AI use

AI systems were used extensively in solving, checking and writing the work. The paper describes AI audits; these do not constitute independent human peer review or formal proof-assistant verification.

Read the paper

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Collaborate on this result

I welcome substantive collaboration on independent verification of the global-action and homology argument, its relationship to the classical decomposition literature, and preparation for journal review. Expertise in post-Lie algebras, Lie-group actions or complex Lie theory would be particularly useful.

Authorship and responsibilities would be discussed in relation to the substantive contributions made.

Cite this paper

Alec Kriebel. (2026). Post-Lie nonexistence with a semisimple target (Version 1.0.0) [Preprint]. Zenodo. https://doi.org/10.5281/zenodo.23414025

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Background and prior work

• Burde and Dekimpe, Pre-Lie and Post-Lie Algebra Structures (2023): Printed p. 2689 formulates the perfect nonsemisimple-source conjecture for every semisimple target, separately from nearby nilpotent-target conjectures.
• Burde, Dekimpe and Monadjem, Post-Lie algebra structures for perfect Lie algebras: The March 8, 2024 author manuscript gives the established transverse-pair formulation in Proposition 2.6 and earlier nonexistence cases in Theorem 3.10 and Propositions 3.11–3.12.
• Burde and Dekimpe, Post-Lie algebra structures and generalized derivations of semisimple Lie algebras: Theorem 3.3 is the prior obstruction for solvable unimodular sources and nonzero semisimple targets.
• Baues, Prehomogeneous Affine Representations and Flat Pseudo-Riemannian Manifolds: Sections 5.4.1–5.4.2 develop the determinant relative invariant and its transitivity consequence for affine-space actions. The current note adapts that mechanism to a holomorphic action on a complex group.
• Etingof, Lie Groups and Lie Algebras (MIT, Spring 2024): Classical integration, Levi decomposition, polar decomposition and solvable-group contractibility supply the geometric inputs.
• Hatcher, Algebraic Topology: Proposition 1.32 and Theorem 3.26 supply the covering-sheet and top-degree homology facts used in the obstruction.