Anti-pre-Novikov algebras, quasi-triangular and factorizable anti-pre-Novikov bialgebras

Abstract

Firstly, we introduce a notion of anti-pre-Novikov algebras as a new framework for decomposing Novikov algebras. Anti-O-operators on Novikov algebras are developed to provide an algebraic framework for constructing anti-pre-Novikov algebras. Secondly, we introduce a notion of anti-pre-Novikov bialgebras as the bialgebra structures corresponding to a double construction of symmetric quasi-Frobenius Novikov algebras, which is characterized by certain matched pairs of Novikov algebras as well as the compatible anti-pre-Novikov algebras. The study of the coboundary case induces the anti-pre-Novikov Yang-Baxter equation (APN-YBE), whose skew-symmetric solutions yield coboundary anti-pre-Novikov bialgebras. The notion of O-operators on anti-pre-Novikov algebras is studied to construct skew-symmetric solutions of the APN-YBE. Thirdly, we investigate quasi-triangular and factorizable anti-pre-Novikov bialgebras as a special class of coboundary anti-pre-Novikov bialgebras. The solutions of the APN-YBE whose symmetric parts are invariant give rise to a quasi-triangular anti-pre-Novikov bialgebra. Moreover, relative Rota-Baxter operators with weights are introduced to demonstrate solutions of the APN-YBE whose symmetric parts are invariant. Finally, we introduce a notion of quadratic Rota-Baxter anti-pre-Novikov algebras, which is one to one correspondence to a factorizable anti-pre-Novikov bialgebra.