L-quadri-algebras

TL;DR

This paper introduces L-quadri-algebras, a new algebraic structure generalizing quadri-algebras with four operations, linking them to Lie algebras and related algebraic concepts.

Contribution

It defines L-quadri-algebras as Lie algebraic analogues of quadri-algebras and explores their connections to Rota-Baxter operators and the classical Yang-Baxter equation.

Findings

L-quadri-algebras generalize quadri-algebras with four operations.

They relate to Lie algebras via the commutator of the sum of operations.

Connections to Rota-Baxter operators and Yang-Baxter equation are established.

Abstract

Quadri-algebras introduced by Aguiar and Loday are a class of remarkable Loday algebras. In this paper, we introduce a notion of L-quadri-algebra with 4 operations satisfying certain generalized left-symmetry, as a Lie algebraic analogue of quadri-algebra such that the commutator of the sum of the 4 operations is a Lie algebra. Any quadri-algebra is an L-quadri-algebra. Moreover, L-quadri-algebras fit into the framework of the relationships between Loday algebras and their Lie algebraic analogues, extending the well known fact that the commutator of an associative algebra is a Lie algebra. We also give the close relationships between L-quadri-algebras and some interesting structures like Rota-Baxter operators, classical Yang-Baxter equation, some bilinear forms satisfying certain conditions.