  Jacobi/Poisson algebras are algebraic counterparts of Jacobi/Poisson
manifolds. We introduce representations of a Jacobi algebra $A$ and Frobenius
Jacobi algebras as symmetric objects in the category. A characterization
theorem for Frobenius Jacobi algebras is given in terms of integrals on Jacobi
algebras. For a vector space $V$ a non-abelian cohomological type object
${\mathcal J}{\mathcal H}^{2} \, (V, \, A)$ is constructed: it classifies all
Jacobi algebras containing $A$ as a subalgebra of codimension equal to ${\rm
dim} (V)$. Representations of $A$ are used in order to give the decomposition
of ${\mathcal J}{\mathcal H}^{2} \, (V, \, A)$ as a coproduct over all Jacobi
$A$-module structures on $V$. The bicrossed product $P \bowtie Q$ of two
Poisson algebras recently introduced by Ni and Bai appears as a special case of
our construction. A new type of deformations of a given Poisson algebra $Q$ is
introduced and a cohomological type object $\mathcal{H}\mathcal{A}^{2}
\bigl(P,\, Q ~|~ (\triangleleft, \, \triangleright, \, \leftharpoonup, \,
\rightharpoonup)\bigl)$ is explicitly constructed as a classifying set for the
bicrossed descent problem for extensions of Poisson algebras. Several examples
and applications are provided.
