Fermionic Novikov bialgebras, fermionic Novikov Yang-Baxter equations and Rota-Baxter operators

Fermionic Novikov algebras are a special class of pre-Lie algebras with anti-commutative right multiplication operators. They correspond to a certain Hamiltonian superoperator in a supervariable. In this paper, we establish a bialgebra theory for fermionic Novikov algebras via the Manin triples approach. Explicitly, we introduce the notion of a fermionic Novikov bialgebra which is equivalent to a Manin triple of fermionic Novikov algebras as well as a certain matched pair of fermionic Novikov algebras. Moreover, we construct a special fermionic Novikov bialgebra, called quasi-triangular, from a solution of the fermionic Novikov Yang-Baxter equation (FNYBE) whose antisymmetric part is invariant. In particular, a symmetric solution of the FNYBE in a fermionic Novikov algebra provide a triangular fermionic Novikov bialgebra, whereas in turn the notion of pre-fermionic Novikov algebras are also introduced to produce the former. As another subclasses of quasi-triangular fermionic Novikov bialgebras, factorizable fermionic Novikov bialgebras lead to a factorization of the underlying fermionic Novikov algebras. Finally, we show that a quadratic Rota-Baxter fermionic Novikov algebra of weight 0 gives rise to a triangular fermionic Novikov bialgebra. Futhermore, there is a one-to-one correspondence between quadratic Rota-Baxter fermionic Novikov algebras of nonzero weights and factorizable fermionic Novikov bialgebras.