A unified algebraic approach to the classical Yang–Baxter equation

By Chengming Bai. 2007

190 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/938ae1d161f9ad8f6bb8bfde4629955b54eb523c

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Citations in the corpus

On integro-differential algebras (2012, background): “Rota-Baxter algebras have found extensive applications in mathematics and physics, including quantum field theory and the classical Yang-Baxter equation [4, 12, 13, 16, 17, 23].”
Matching Rota-Baxter algebras, matching dendriform algebras and matching pre-Lie algebras (2019, background): “Further, it was established a long time ago that Rota-Baxter operator on a Lie algebra is precisely the operator form of the classical Yang-Baxter equation [5, 39].”