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The matching algebra for the momentum-dependent parts is known only for a simplified self-dual sector of the theory.","posed_since":"2011","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.","problem_ref":null,"references":"","settled_by":"An explicit algebra reproducing tree-level Yang-Mills numerators at all multiplicities, or a no-go proof.","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.","title":"Kinematic Lie algebra of full Yang-Mills 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