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 "O-operator", whereas a skew-symmetric solution of the classical Yang-Baxter equation corresponds to the skew-symmetric part of an O-operator. Thus a method to construct symmetric solutions of the S-equation (hence parakahler Lie algebras) from 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.