Pseudo-Hessian Lie algebras and L-dendriform bialgebras

In this paper, we study a special class of pseudo-Hessian Lie algebras satisfying an additional condition that they are decomposed into a direct sum of underlying vector spaces of two Lagrangian subalgebras in terms of L-dendriform algebras which are the underlying algebraic structures. Such structures are equivalent to certain bialgebra structures, namely, L-dendriform bialgebras. Furthermore, we introduce and study the so-called triangular pseudo-Hessian Lie algebras and relate them to some semidirect product constructions from the "Lie-type" operations of L-dendriform algebras.