  Given a Lie algebra, there uniquely exists a Poisson algebra which is called
a Lie-Poisson algebra over the Lie algebra. We will prove that given a
Loday/Leibniz algebra there exists uniquely a noncommutative Poisson algebra
over the Loday algebra. The noncommutative Poisson algebras are called the
Loday-Poisson algebras. In the super/graded cases, the Loday-Poisson bracket is
regarded as a noncommutative version of classical (linear) Schouten-Nijenhuis
bracket. It will be shown that the Loday-Poisson algebras form a special
subclass of Aguiar's dual-prePoisson algebras. We also study a problem of
deformation quantization over the Loday-Poisson algebra. It will be shown that
the polynomial Loday-Poisson algebra is deformation quantizable and that the
associated quantum algebra is Loday's associative dialgebra.
