A Unified Algebraic Approach to Classical Yang-Baxter Equation

Details A Unified Algebraic Approach to Classical Yang-Baxter Equation A Unified Algebraic Approach to Classical Yang-Baxter Equation

2007-07-28

arxiv.org/abs/0707.4226v1

Journal of Physics A: Mathematical and Theoretical 40 (2007) 11073-11082

Mathematics

Quantum Algebra

To appear in Journal of Physics A: Mathematical and Theoretical

Scientific paper

10.1088/1751-8113/40/36/007

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.

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