  Attention is focused on q-deformed quantum algebras with physical importance,
i.e. $U_{q}(su_{2})$, $U_{q}(so_{4})$ and q-deformed Lorentz algebra. The main
concern of this article is to assemble important ideas about these symmetry
algebras in a consistent framework which shall serve as starting point for
representation theoretic investigations in physics, especially quantum field
theory. In each case considerations start from a realization of symmetry
generators within the differential algebra. Formulae for coproducts and
antipodes on symmetry generators are listed. The action of symmetry generators
in terms of their Hopf structure is taken as q-analog of classical commutators
and written out explicitly. Spinor and vector representations of symmetry
generators are calculated. A review of the commutation relations between
symmetry generators and components of a spinor or vector operator is given.
Relations for the corresponding quantum Lie algebras are computed. Their
Casimir operators are written down in a form similar to the undeformed case.
