Kinematic Lie algebra of full Yang-Mills theory
In plain words
Color factors come from a known algebra, the rules for combining color charges. The matching algebra for the momentum-dependent parts is known only for a simplified self-dual sector of the theory.
Precise statement
Identify an infinite-dimensional Lie algebra with structure constants $F$ such that tree-level BCJ numerators of $D$-dimensional Yang-Mills at all multiplicities are nested products of F with the same graph structure as nested color factors $f^{abc}$. In the self-dual sector the algebra is that of area-preserving diffeomorphisms (Monteiro and O'Connell, 2011); an answer is the algebra for the full theory, or a proof that no Lie algebra (as opposed to a Hopf, homotopy or other structure) can do this.
What would settle it
An explicit algebra reproducing tree-level Yang-Mills numerators at all multiplicities, or a no-go proof.
Status in the literature
Unverified note
Kinematic Hopf-algebra constructions (from 2021) generate tree-level BCJ numerators, and homotopy-algebra work (Bonezzi, Chiaffrino, Diaz-Jaramillo, Hohm, arXiv:2212.04513, 2023) finds a generalized Batalin-Vilkovisky algebra in off-shell Yang-Mills; no strict Lie algebra for the full theory is known as of 2026.