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Title: Kinematic Lie algebra of full Yang-Mills theory
Plain: 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: 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.
Settled by: An explicit algebra reproducing tree-level Yang-Mills numerators at all multiplicities, or a no-go proof.
Kind: well-posed
Answer type: identification
References: (empty)
Literature status: partially-resolved
Status 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.
Posed since: 2011