Extended $\mathcal{O}$-operators, Novikov Yang-Baxter equations and post-Novikov algebras
Jianfeng Yu, Yanyong Hong
TL;DR
This work develops an operator-theoretic framework for Novikov algebras by introducing extended $\mathcal{O}$-operators associated to $A$-bimodule Novikov algebras. It builds a bridge from these operators to post-Novikov algebras and to new Novikov algebra structures, and then elevates the study to tensor forms yielding extended Novikov Yang-Baxter equations (ENYBE) and generalized NYBE (GNYBE). The authors establish precise correspondences between extended $\mathcal{O}$-operators and (extended) NYBE, including special cases such as Rota–Baxter operators, Baxter-Novikov algebras, and invariant bilinear forms on quadratic Novikov algebras. The results unify multiple strands of Novikov bialgebra theory and provide operator-based tools for constructing and analyzing Novikov algebraic structures and their deformations.
Abstract
In this paper, we introduce the definition of extended $\mathcal{O}$-operators on a Novikov algebra $(A,\circ)$ associated to an $A$-bimodule Novikov algebra which is a generalization of the definition of $\mathcal{O}$-operators and show that there are new Novikov algebra structures on the $A$-bimodule Novikov algebra obtained from extended $\mathcal{O}$-operators. We also introduce the definition of post-Novikov algebras and show that there is a close relationship between post-Novikov algebras and $\mathcal{O}$-operators of weight $λ$. The tensor form of extended $\mathcal{O}$-operators is also investigated which leads to the definition of extended Novikov Yang-Baxter equations, which is a generalization of the notion of Novikov Yang-Baxter equations. The relationships between extended $\mathcal{O}$-operators, Novikov Yang-Baxter equations, extended Novikov Yang-Baxter equations and generalized Novikov Yang-Baxter equations are established.