Noncommutative Novikov bialgebras and differential antisymmetric infinitesimal bialgebras with weight

Abstract

This paper first develops a bialgebra theory for a noncommutative Novikov algebra, called a noncommutative Novikov bialgebra, which is further characterized by matched pairs and Manin triples of noncommutative Novikov algebras. The classical Yang-Baxter type equation, $\mathcal{O}$-operators, and noncommutative pre-Novikov algebras are introduced to study noncommutative Novikov bialgebra. As an application, noncommutative pre-Novikov algebras are obtained from differential dendriform algebras. Next, to generalize Gelfand's classical construction of a Novikov algebra from a commutative differential algebra to the bialgebra context in the noncommutative case, we establish antisymmetric infinitesimal (ASI) bialgebras for (noncommutative) differential algebras, and obtain the condition under which a differential ASI bialgebra induces a noncommutative Novikov bialgebra.