Noncommutative Novikov bialgebras and differential antisymmetric infinitesimal bialgebras with weight
Shanghua Zheng, Yizhen Li, Liushuting Yang, Li Guo
TL;DR
The paper develops a comprehensive bialgebra framework for noncommutative Novikov and differential algebras, extending Gelfand's classical commutative construction to the noncommutative setting via Manin triples and matched pairs. It introduces the Novikov Yang–Baxter equation (NYBE) and $\mathcal{O}$-operators as main tools, and shows how noncommutative pre-Novikov algebras arise from differential dendriform algebras and provide constructive NYBE solutions. It then advances differential antisymmetric infinitesimal (ASI) bialgebras of weight $\lambda$, giving representations, double constructions, and characterizations; crucially, weight-0 differential ASI bialgebras yield noncommutative Novikov bialgebras under compatibility. The work unifies triangular and quasi-Frobenius structures, dualities, and derived structures across bialgebras, Frobenius, and differential settings, enriching the algebraic toolkit for noncommutative Novikov-type theories and their potential applications.
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.