Efficient Normalizers of Non-Chevalley Derivation Lie Algebras

Definitions and mathematical statement

For a field $k$ and a $k$-algebra $A$ with bilinear multiplication, its derivation Lie algebra is \[ \operatorname{Der}_k(A)=\{D\in\operatorname{End}_k(A):D(ab)=D(a)b+aD(b)\text{ for all }a,b\in A\}, \] with bracket $[D,E]=D\circ E-E\circ D$. This definition explains the usual algebra interpretation of derivations; the supplied record does not specify the objects whose derivations are intended, so it does not establish that this interpretation exhausts the problem's scope. A Lie algebra is called simple here if it is nonzero, nonabelian, and has no nonzero proper ideals. The intended meaning of "non-Chevalley" is not defined in the supplied record and must be fixed before selecting the input class.

For a represented Lie algebra $L\subseteq\mathfrak{gl}_k(V)=\operatorname{End}_k(V)$ and a specified subgroup $G\leq\operatorname{GL}_k(V)$, the group normalizer is \[ N_G(L)=\{g\in G:gLg^{-1}=L\}. \] A different possible target is the Lie algebra normalizer \[ N_{\mathfrak{gl}_k(V)}(L)=\{X\in\mathfrak{gl}_k(V):[X,L]\subseteq L\}. \] These definitions describe distinct possible readings; the source record does not select an ambient group, an ambient Lie algebra, or a representation.

Source question. Can normalizers of the non-Chevalley simple Lie algebras that arise as derivation algebras be constructed efficiently?

Schematically, the requested construction should apply to every member of the intended class of non-Chevalley simple derivation Lie algebras. However, that class, its ground fields and characteristics, its representations, the ambient normalizer, and the computational input and output conventions remain unspecified. In particular, "efficiently" cannot yet be replaced by a precise running-time bound. The question is therefore retained as an unresolved algorithmic formulation, rather than strengthened to a polynomial-time assertion or restricted to an invented domain.

Short English statement

The problem asks for an efficient method to construct normalizers of non-Chevalley simple Lie algebras occurring as derivation algebras. A precise algorithmic statement requires the intended fields, derivation objects, representations, ambient normalizers, and meaning of efficiency. The supplied registry labels the problem open; this draft does not independently verify that status.

Sources

• 1. Supplied local UnsolvedMath registry record, permanent ID $30004766$, title "Efficient Normalizers of Non-Chevalley Derivation Lie Algebras," exact source problem code catalogue entry, in the collection "Oberwolfach Reports Open Problems." The record gives proposed year $2021$ and names no individual proposer.
• 2. Supplied report citation: Computational Group Theory (2022), Oberwolfach Reports, DOI 10.4171/owr/2021/38, pp. 2027--2087. https://doi.org/10.4171/owr/2021/38. The literature-source label is "Oberwolfach Report 38/2021." The supplied citation's $2022$ designation and the report's $2021$ designation are retained without reconciliation.
• 3. Supplied literature source: Tensor Isomorphism by Derivations and Densors. https://arxiv.org/abs/2005.04046. The supplied record does not identify authors or a specific version.
• 4. Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and reports that the cited work provides efficient normalizer constructions for important Chevalley-type cases while leaving the requested construction for non-Chevalley simple Lie algebras arising as derivation algebras as a future problem. This is supplied dated curation, not independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• 5. The supplied registry assigns statement status corrected_verified and reports that the official Oberwolfach report was checked directly and that the clean statement preserves the complete mathematical target while restoring definitions and removing extraction artifacts or source-dependent wording. These are supplied verification claims; no independent source consultation is asserted here.