  The FRT quantum group and space theory is reformulated from the standard
mathematical basis to an arbitrary one. The $N$-dimensional quantum vector
Cayley-Klein spaces are described in Cartesian basis and the quantum analogs of
$(N-1)$-dimensional constant curvature spaces are introduced. Part of the
4-dimensional constant curvature spaces are interpreted as the non-commutative
analogs of $(1+3)$ kinematics. A different unifications of Cayley-Klein and
Hopf structures in a kinematics are described with the help of permutations.
All permutations which lead to the physically nonequivalent kinematics are
found and the corresponding non-commutative $(1+3)$ kinematics are
investigated. As a result the quantum (anti) de Sitter, Minkowski, Newton,
Galilei kinematics with the fundamental length, the fundamental mass and the
fundamental velocity are obtained.
