Quasi-triangular Novikov bialgebras and related bialgebra structures
Zhanpeng Cui, Bo Hou
TL;DR
This work develops a comprehensive framework for quasi-triangular and factorizable Novikov bialgebras, connecting solutions of the NYBE to structural bialgebra properties and their duals. It establishes precise correspondences with quadratic Rota-Baxter Novikov algebras, and shows how these notions lift from associative and differential settings to Novikov and Lie bialgebras, including double constructions and induced Lie bialgebras via quadratic right Novikov algebras. The results provide systematic methods to generate quasi-triangular and factorizable structures from differential infinitesimal bialgebras and from the Novikov double, with explicit criteria involving invariance of symmetric parts and bijectivity of key linear maps. Overall, the work extends the interplay between NYBE, AYBE, and CYBE in the Novikov realm and enhances tools for constructing and classifying related Lie bialgebras.
Abstract
We introduce the notion of quasi-triangular Novikov bialgebras, which constructed from solutions of the Novikov Yang-Baxter equation whose symmetric parts are invariant. Triangular Novikov bialgebras and factorizable Novikov bialgebras are important subclasses of quasi-triangular Novikov bialgebras. A factorizable Novikov bialgebra induces a factorization of the underlying Novikov algebra and the double of any Novikov bialgebra naturally admits a factorizable Novikov bialgebra structure. Moreover, we introduce the notion of quadratic Rota-Baxter Novikov algebras and show that there is an one-to-one correspondence between factorizable Novikov bialgebras and quadratic Rota-Baxter Novikov algebras of nonzero weights. Finally, we obtain that the Lie bialgebra induced by a Novikov bialgebra and a quadratic right Novikov algebra is quasi-triangular (resp. triangular, factorizable) if the Novikov bialgebra is quasi-triangular (resp. triangular, factorizable), and under certain conditions, the Novikov bialgebra induced by a differential infinitesimal bialgebra is quasi-triangular (resp. triangular, factorizable) if the differential infinitesimal bialgebra is quasi-triangular (resp. triangular, factorizable).