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

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 parakahler 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 parakahler Lie algebra. We also show that such a solution corresponds to the symmetric part of a certain operator called ${cal O}$-operator, whereas a skew-symmetric solution of the classical Yang-Baxter equation corresponds to the skew-symmetric part of an ${cal O}$-operator. Thus a method to construct symmetric solutions of the $S$-equation (hence parakahler Lie algebras) from ${cal O}$-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, parakahler Lie groups correspond to Poisson-Lie groups in this sense.