Infinite-dimensional pre-Lie bialgebras via affinization of pre-Novikov bialgebras

In this paper, we show that there is a pre-Lie algebra structure on the tensor product of a pre-Novikov algebra and a right Novikov dialgebra and the tensor product of a pre-Novikov algebra and a special right Novikov algebra on the vector space of Laurent polynomials being a pre-Lie algebra characterizes the pre-Novikov algebra. The latter is called the affinization of a pre-Novikov algebra. Moreover, we extend this construction of pre-Lie algebras and the affinization of pre-Novikov algebras to the context of bialgebras. We show that there is a completed pre-Lie bialgebra structure on the tensor product of a pre-Novikov bialgebra and a quadratic Z-graded right Novikov algebra. Moreover, a pre-Novikov bialgebra can be characterized by the fact that its affinization by a quadratic Z-graded right Novikov algebra on the vector space of Laurent polynomials gives an infinite-dimensional completed pre-Lie bialgebras. Note that the reason why we choose a quadratic right Novikov algebra instead of a right Novikov dialgebra with a special bilinear form is also given. Furthermore, we construct symmetric completed solutions of the S-equation in the induced pre-Lie algebra by symmetric solutions of the pre-Novikov Yang-Baxter equation in a pre-Novikov algebra.