Some results on L-dendriform algebras

TL;DR

This paper introduces L-dendriform algebras, exploring their role in pseudo-Hessian structures, $ ext{O}$-operators, and solutions to algebraic equations analogous to the Yang-Baxter equation, expanding the algebraic framework.

Contribution

It defines L-dendriform algebras, links them to various algebraic structures, and introduces $ ext{O}$-operators, providing new insights into algebraic equations similar to the Yang-Baxter equation.

Findings

L-dendriform algebras underpin pseudo-Hessian structures on Lie groups.

Explicit solutions to the $S$-equation are derived from L-dendriform algebras.

A new $ ext{O}$-operator concept leads to an algebraic equation analogous to the classical Yang-Baxter equation.

Abstract

We introduce a notion of 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 $\mathcal O$-operators of pre-Lie algebras and the related $S$-equation. As a direct consequence, they provide some explicit solutions of $S$-equation 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 a notion of $\mathcal O$-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.