  We introduce the notion of a braided algebra and study some examples of
these. In particular, R-symmetric and R-skew-symmetric algebras of a linear
space V equipped with a skew-invertible Hecke symmetry R are braided algebras.
We prove the "mountain property" for the numerators and denominators of their
Poincare-Hilbert series (which are always rational functions).
  Also, we further develop a differential calculus on modified Reflection
Equation algebras. Thus, we exhibit a new form of the Leibniz rule for partial
derivatives on such algebras related to involutive symmetries R. In particular,
we present this rule for the algebra U(gl(m)). The case of the algebra U(gl(2))
and its compact form U(u(2)) (which can be treated as a deformation of the
Minkowski space algebra) is considered in detail. On the algebra U(u(2)) we
introduce the notion of the quantum radius, which is a deformation of the usual
radius, and compute the action of rotationally invariant operators and in
particular of the Laplace operator. This enables us to define analogs of the
Laplace-Beltrami operators corresponding to certain Schwarzschild-type metrics
and to compute their actions on the algebra U(u(2)) and its central extension.
Some "physical" consequences of our considerations are presented.
