Noncommutative Novikov bialgebras and differential antisymmetric infinitesimal bialgebras via manin triples

This paper first establishes a bialgebra structure for a noncommutative Novikov algebra, called a noncommutative Novikov bialgebra, which is characterized by matched pairs and Manin triples of noncommutative Novikov algebras. The classical Yang-Baxter type equation, O-operators, and noncommutative pre-Novikov algebras are introduced to study noncommutative Novikov bialgebras. As an application, noncommutative pre-Novikov algebras are derived from differential dendriform algebras. Furthermore, to lift Gelfand's classical construction of a Novikov algebra from a commutative differential algebra to the level of bialgebras in the noncommutative context, we establish antisymmetric infinitesimal (ASI) bialgebras for (noncommutative) differential algebras, and obtain the condition under which a differential ASI bialgebra yields a noncommutative Novikov bialgebra.