  Generalizing S. Gelfand's classical construction of a Novikov algebra from a
commutative differential algebra, a deformation family $(A,\circ_q)$, 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 $(A,\circ_0)$ 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
$(A,\circ_{-\frac{1}{2}})$ instead of $(A,\circ_0)$. The close relations of the
classical bialgebra theories with Manin triples, classical Yang-Baxter type
equations, $\mathcal{O}$-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.
