  A bicovariant calculus of differential operators on a quantum group is
constructed in a natural way, using invariant maps from \fun\ to \uqg\ , given
by elements of the pure braid group. These operators --- the `reflection
matrix' $Y \equiv L^+ SL^-$ being a special case --- generate algebras that
linearly close under adjoint actions, i.e. they form generalized Lie algebras.
We establish the connection between the Hopf algebra formulation of the
calculus and a formulation in compact matrix form which is quite powerful for
actual computations and as applications we find the quantum determinant and an
orthogonality relation for $Y$ in $SO_q(N)$.
