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Colour-kinematics duality suggests that Yang-Mills (YM) theory possesses somehidden Lie algebraic structure. So far this structure has resistedunderstanding, apart from some progress in the self-dual sector. We show thatthere is indeed a Lie algebra behind the YM Feynman rules. The Lie algebra weuncover is the Drinfeld double of the Lie algebra of vector fields. Morespecifically, we show that the kinematic numerators following from the YMFeynman rules satisfy a version of the Jacobi identity, in that the Jacobiatorof the bracket defined by the YM cubic vertex is cancelled by the contributionof the YM quartic vertex. We then show that this Jacobi-like identity is infact the Jacobi identity of the Drinfeld double. All our considerations areoff-shell. Our construction explains why numerators computed using the Feynmanrules satisfy the colour-kinematics at four but not at higher numbers ofpoints. It also suggests a way of modifying the Feynman rules so that theduality can continue to hold for an arbitrary number of gluons. Ourconstruction stops short of producing explicit higher point numerators becauseof an absence of a certain property at four points. We comment on possible waysof correcting this, but leave the next word in the story to future work.

Colour-Kinematics duality and the Drinfeld double of the Lie algebra of diffeomorphisms