Post-Lie Structures with Semisimple Target

Definitions and mathematical statement

For a finite-dimensional complex vector space $V$, let $[\ ,\ ]$ and $\{\ ,\ \}$ be Lie brackets on $V$, defining $\mathfrak g=(V,[\ ,\ ])$ and $\mathfrak n=(V,\{\ ,\ \})$. A Lie algebra $\mathfrak a$ is perfect if $[\mathfrak a,\mathfrak a]=\mathfrak a$, where the derived algebra is the linear span of its brackets. In the finite-dimensional setting, $\mathfrak a$ is semisimple if its solvable radical, the largest solvable ideal, is zero. Here an ideal $I$ is solvable if its derived series $I^{(0)}=I$, $I^{(k+1)}=[I^{(k)},I^{(k)}]$ reaches zero.

A post-Lie algebra structure on the ordered pair $(\mathfrak g,\mathfrak n)$ is a complex-bilinear product $\cdot:V\times V\to V$ satisfying, for all $x,y,z\in V$, \[ \begin{aligned} x\cdot y-y\cdot x&=[x,y]-\{x,y\},\\ [x,y]\cdot z&=x\cdot(y\cdot z)-y\cdot(x\cdot z),\\ x\cdot\{y,z\}&=\{x\cdot y,z\}+\{y,x\cdot z\}. \end{aligned} \]

Conjecture, under the finite-dimensional interpretation. For every finite-dimensional complex vector space $V$ and every pair of Lie brackets on $V$, if $\mathfrak g$ is perfect and nonsemisimple and $\mathfrak n$ is semisimple, then no such post-Lie product exists.

Unresolved scope qualification. The supplied statement does not explicitly impose finite dimensionality. The formulation above records the standard finite-dimensional interpretation as a provisional scope choice requiring human confirmation; it does not assert an extension to infinite-dimensional Lie algebras. The shared underlying vector space is made explicit to define the post-Lie identities. The supplied original excerpt refers to an unspecified preceding conclusion, so the nonexistence conclusion and the nonsemisimplicity hypothesis on $\mathfrak g$ are taken from the supplied reconstructed statement.

Short English statement

The supplied conjecture says that a perfect, nonsemisimple complex Lie algebra cannot admit a post-Lie product with a semisimple second Lie bracket on the same vector space. The displayed formulation uses a provisional finite-dimensional interpretation because the supplied record leaves the dimensional scope unstated. The record labels the problem open and reports that the case $\mathfrak n\cong\mathfrak{sl}_3({\mathbb{C}})$ is proved; these are supplied assessments requiring independent review.

Sources

• Local UnsolvedMath registry, permanent ID $30005737$, source problem code catalogue entry, title Post-Lie Structures with Semisimple Target; supplied collection: Oberwolfach Reports Open Problems. No individual proposer is identified; the supplied proposed year is 2023.
• Supplied source citation: Mini-Workshop: Poisson and Poisson-type algebras (2024), Oberwolfach Reports, pp. 2681--2715, DOI 10.4171/owr/2023/46. Supplied literature-source label: Oberwolfach Report 46/2023. https://doi.org/10.4171/owr/2023/46. The supplied original excerpt points to p. 2689 and conjectures an extension from $\mathfrak{sl}_3({\mathbb{C}})$ to arbitrary semisimple $\mathfrak n$.
• Supplied literature assessment, checked on 2026-08-22: the report is said to prove only the $\mathfrak{sl}_3({\mathbb{C}})$ case and to present the general semisimple case as a conjecture; the supplied assessment says no later resolving proof was located. This is supplied dated curation, not an independently verified literature assessment. Its listed source is https://doi.org/10.4171/owr/2023/46.
• The registry labels the statement corrected_verified and says it was reconstructed and checked against the official report. These are supplied provenance claims; the report has not been independently consulted for this draft.