On left-symmetric conformal bialgebras

The notion of left-symmetric bialgebra was introduced in [C. M. Bai, Left-symmetric bialgebra and an analogue of the classical Yang–Baxter equation, Commun. Contemp. Math. 10(2) (2008) 221–260] which is equivalent to a parakähler Lie algebra which is the Lie algebra of a Lie group G with a G-invariant parakähler structure. In this paper, we study a conformal analog of left-symmetric bialgebras. The notions of left-symmetric conformal coalgebra and bialgebra are introduced. Moreover, the constructions of matched pairs of Lie conformal algebras and left-symmetric conformal algebras are presented. We show that a finite left-symmetric conformal bialgebra which is free as a ℂ[∂]-module is equivalent to a parakähler Lie conformal algebra. We also obtain a conformal analog of the S-equation (see [C. M. Bai, Left-symmetric bialgebras and an analogue of the classical Yang–Baxter equation, Commun. Contemp. Math. 10(2) (2008) 221–260]), and give a construction of the conformal symplectic double.