A Unified Algebraic Approach to Classical Yang-Baxter Equation

2007/07/28 by Chengming Bai · 2 citations
Mathematics · Physics and Astronomy · #math.QA #math-ph #math.MP #msc:17Bxx #msc:81Rxx

paper · pdf · doi:10.1088/1751-8113/40/36/007

published as Journal of Physics A: Mathematical and Theoretical 40 (2007) 11073-11082 · To appear in Journal of Physics A: Mathematical and Theoretical

arxiv created 2007/07/28 · arxiv updated 2009/12/01

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 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.

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• Hom-Poisson superalgebras with nondegenerate bilinear forms and compatible Hom-anti-pre-Poisson superalgebras
• Manin triples, bialgebras and Yang-Baxter equation of A₃-associative algebras

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