  We consider new class of classical r-matrices for D=3 and D=4 conformal Lie
algebras. These r-matrices do satisfy the classical Yang-Baxter equation and as
two-tensors belong to the tensor product of Borel subalgebra. In such a way we
generalize the lowest order of known nonstandard quantum deformation of sl(2)
to the Lie algebras sp(4) = so(5) and sl(4) = so(6). As an exercise we
interpret nonstandard deformation of sl(2) as describing quantum D=1 conformal
algebra with fundamental mass parameter. Further we describe the D=3 and D=4
conformal bialgebras with deformation parameters equal to the inverse of
fundamental masses. It appears that for D=4 the deformation of the Poincare
algebra sector coincides with "null plane" quantum Poincare algebra.
