Left-symmetric Bialgebras and An Analogue of the Classical Yang-Baxter Equation

Abstract

We introduce a notion of left-symmetric bialgebra which is an analogue of the notion of Lie bialgebra. We prove that a left-symmetric bialgebra is equivalent to a symplectic Lie algebra with a decomposition into a direct sum of the underlying vector spaces of two Lagrangian subalgebras. The latter is called a parakähler Lie algebra or a phase space of a Lie algebra in mathematical physics. We introduce and study coboundary left-symmetric bialgebras and our study leads to what we call "$S$-equation", which is an analogue of the classical Yang-Baxter equation. In a certain sense, the $S$-equation associated to a left-symmetric algebra reveals the left-symmetry of the products. We show that a symmetric solution of the $S$-equation gives a parakähler Lie algebra. We also show that such a solution corresponds to the symmetric part of a certain operator called "$\mathcal{O}$-operator", whereas a skew-symmetric solution of the classical Yang-Baxter equation corresponds to the skew-symmetric part of an $\mathcal{O}$-operator. Thus a method to construct symmetric solutions of the $S$-equation (hence parakähler Lie algebras) from $\mathcal{O}$-operators is provided. Moreover, by comparing left-symmetric bialgebras and Lie bialgebras, we observe that there is a clear analogue between them and, in particular, parakähler Lie groups correspond to Poisson-Lie groups in this sense.

arXiv comment44 pages

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• Conformal classical Yang-Baxter equation, $S$-equation and $\mathcal{O}$-operators
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• Anti-flexible bialgebras
• Categorification of Pre-Lie Algebras and Solutions of 2-graded Classical Yang-Baxter Equations
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• Review of deformation theory I: Concrete formulas for deformations of algebraic structures
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• Cohomology and Deformations of left-symmetric Rinehart Algebras
• Koszul-Vinberg structures and compatible structures on left-symmetric algebroids
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• Deformations and Cohomologies of Relative Rota-Baxter Operators on Lie Algebroids and Koszul-Vinberg Structures
• Twisting on pre-Lie algebras and quasi-pre-Lie bialgebras
• Extending structures and classifying complements for left-symmetric algebras
• On $k$-para-Kähler Lie algebras a subclass of $k$-symplectic Lie algebras
• Pre-Lie analogues of Poisson-Nijenhuis structures and Maurer-Cartan equations
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