Hopf stars, twisted Hopf stars and scalar products on quantum spaces

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.