Some results on L-dendriform algebras

We introduce the notion of an L-dendriform algebra due to several different motivations. L-dendriform algebras are regarded as the underlying algebraic structures of pseudo-Hessian structures on Lie groups and the algebraic structures behind the OO-operators of pre-Lie algebras and the related SS-equation. As a direct consequence, they provide some explicit solutions of SS-equations in certain pre-Lie algebras constructed from L-dendriform algebras. They also fit into a bigger framework as Lie algebraic analogues of dendriform algebras. Moreover, we introduce the notion of an OO-operator of an L-dendriform algebra which gives an algebraic equation regarded as an analogue of the classical Yang–Baxter equation in a Lie algebra.