Gauge independent kinematic algebra of self-dual Yang-Mills theory

The double-copy program relies crucially on the so-called color-kinematics duality which, in turn, is widely believed to descend from a kinematic algebra possessed by gauge theories. In this paper we construct the kinematic algebra of gauge-invariant and off shell self-dual Yang-Mills theory, up to trilinear maps. This structure is a homotopy algebra of the same type as the ones recently uncovered in Chern-Simons and full Yang-Mills theories. To make contact with known results for the self-dual sector, we show that it reduces to the algebra found by Monteiro and O'Connell upon taking the light cone gauge and partially solving the self-duality constraints. Finally, we test a double-copy prescription recently proposed in Bonezzi et al. [Gauge invariant double copy of Yang-Mills theory: The quartic theory, Phys. Rev. D 107, 126015 (2023), 10.1103/PhysRevD.107.126015] and reproduce self-dual gravity.