Abstract
We study dual binary Zinbiel algebras over a field of characteristic zero, defined as the quadratic duals of binary Zinbiel algebras. We first analyze the multilinear component of degree three of the free binary Zinbiel algebra and construct an explicit basis consisting of seven elements. Using this basis, we determine the defining identities of the dual operad. We prove that a dual binary Zinbiel algebra is characterized by the Leibniz identity together with the identity $a(aa)=0$. Thus, the quadratic dual of the binary Zinbiel operad is described by two simple identities. We then investigate the Koszulity of the binary Zinbiel operad. Using the first five multilinear dimensions of the operad and its quadratic dual, we compute the corresponding exponential generating series. Substitution into the Ginzburg–Kapranov functional equation shows that the required identity fails in degree five. Consequently, the binary Zinbiel operad is not Koszul, and neither is its quadratic dual.