A unified algebraic approach to the classical Yang–Baxter equation

Abstract

In this paper, the different operator forms of the 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 the 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 parakähler 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.