A Unified Algebraic Approach to Classical Yang-Baxter Equation

TL;DR

This paper presents a unified algebraic framework for the classical Yang-Baxter equation, linking it to left-symmetric algebras and geometric structures, and exploring their interrelations.

Contribution

It introduces a unified tensor expression approach and establishes new connections between classical r-matrices and left-symmetric algebras.

Findings

Unified algebraic method for classical Yang-Baxter equation

Construction of left-symmetric algebras from classical r-matrices

Correspondence between left-symmetric algebras and parakähler structures

Abstract

In this paper, the different operator forms of classical Yang-Baxter equation are given in the tensor expression through a unified algebraic method. It is closely related to left-symmetric algebras which play an important role in many fields in mathematics and mathematical physics. By studying the relations between left-symmetric algebras and classical Yang-Baxter equation, we can construct left-symmetric algebras from certain classical r-matrices and conversely, there is a natural classical r-matrix constructed from a left-symmetric algebra which corresponds to a parak\"ahler structure in geometry. Moreover, the former in a special case gives an algebraic interpretation of the ``left-symmetry'' as a Lie bracket ``left-twisted'' by a classical r-matrix.