Deformation families of Novikov bialgebras via differential antisymmetric infinitesimal bialgebras

Generalizing S. Gelfand's classical construction of a Novikov algebra from a commutative differential algebra, a deformation family (A,∘_q), for scalars q, of Novikov algebras is constructed from what we call an admissible commutative differential algebra, by adding a second linear operator to the commutative differential algebra with certain admissibility condition. The case of (A,∘_0) recovers the construction of S. Gelfand. This admissibility condition also ensures a bialgebra theory of commutative differential algebras, enriching the antisymmetric infinitesimal bialgebra. This way, a deformation family of Novikov bialgebras is obtained, under the further condition that the two operators are bialgebra derivations. As a special case, we obtain a bialgebra variation of S. Gelfand's construction with an interesting twist: every commutative and cocommutative differential antisymmetric infinitesimal bialgebra gives rise to a Novikov bialgebra whose underlying Novikov algebra is (A,∘_-1/2) instead of (A,∘_0). The close relations of the classical bialgebra theories with Manin triples, classical Yang-Baxter type equations, 𝒪-operators, and pre-structures are expanded to the two new bialgebra theories, in a way that is compatible with the just established connection between the two bialgebras. As an application, Novikov bialgebras are obtained from admissible differential Zinbiel algebras.