The operad of anti-pre-Lie algebras and its Lie elements

We initiate a systematic operadic approach to anti-pre-Lie algebras. We study the operad controlling anti-pre-Lie algebras, its relationship to the operad of Novikov algebras and its cyclic structure. We establish that the Lie bracket of an anti-pre-Lie algebra satisfies the standard identity of degree five. We also establish a quadratic maximality property of the anti-pre-Lie operad with respect to this identity. We then show that the anti-pre-Lie operad and its Koszul dual are not Koszul and give a full description of the latter. Finally, we identify the Lie elements in the Koszul dual operad with the operad of metabelian Lie algebras.