Quasi-triangular Novikov bialgebras and related bialgebra structures

TL;DR

This paper introduces quasi-triangular Novikov bialgebras, explores their subclasses, and establishes connections with quadratic Rota-Baxter Novikov algebras, expanding the understanding of Novikov bialgebra structures.

Contribution

It defines quasi-triangular Novikov bialgebras, relates them to factorizable and triangular subclasses, and links them to quadratic Rota-Baxter Novikov algebras, providing new structural insights.

Findings

Factorizable Novikov bialgebras induce algebra factorizations.

A one-to-one correspondence exists between factorizable Novikov bialgebras and quadratic Rota-Baxter Novikov algebras.

Induced Lie bialgebras inherit quasi-triangular, triangular, or factorizable structures under certain conditions.

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).