  The properties of Hopf star operations and twisted Hopf stars operations on
quantum groups are discussed in relation with the theory of representations
(star representations). Invariant Hermitian sesquilinear forms (scalar
products) on modules or module-algebras are then defined and analyzed.
Particular attention is paid to scalar products that can be associated with the
Killing form (when it exists) or with the left (or right) invariant integrals
on the quantum group.
  Our results are systematically illustrated in the case of a family of non
semi-simple and finite dimensional quantum groups that are obtained as Hopf
quotients of the quantum enveloping algebra U_q(sl(2,C)), q being an N-th root
of unity. Many explicit results concerning the case N=3 are given.
  We also mention several physical motivations for the present work: conformal
field theory, spin chains, integrable models, generalized Yang-Mills theory
with quantum group action and the search for finite quantum groups symmetries
in particle physics.
