The Levi-Civita products of Leibniz algebras with nondegenerate skew-symmetric 2-cocycles

Abstract

This paper studies the associated Levi-Civita products of a Leibniz algebra with a nondegenerate skew-symmetric $2$-cocycle. Such products form into the notion of an anti-pre-Leibniz algebra, which is characterized as a Leibniz-admissible algebra which renders a representation of the sub-adjacent Leibniz algebra through the negative multiplication operators. Such a characterization serves as the converse side of the role that pre-Liebniz algebras play in the splitting theory of Leibniz algebras, which justifies the name of anti-pre-Leibniz algebras. There is a compatible anti-pre-Leibniz algebra structure on a Leibniz algebra if and only if there is an invertible anti-$\mathcal{O}$-operator of the Leibniz algebra. Another important role that anti-pre-Leibniz algebras play is that they give a new characterization of Novikov dialgebras, that is, a Novikov dialgebra is interpreted as a transformed pre-Leibniz algebra which gives rise to an anti-pre-Leibniz algebra structure through specific combinations of multiplications. The properties of Novikov dialgebras are also further investigated.