The machinery of braided geometry introduced previously is used now to construct the $\epsilon$ `totally antisymmetric tensor' on a general braided vector space determined by R-matrices. This includes natural $q$-Euclidean and $q$-Minkowski spaces. The formalism is completely covariant under the corresponding quantum group such as $\widetilde{ SO_q(4)}$ or $\widetilde{ SO_q(1,3)}$. The Hodge $*$ operator and differentials are also constructed in this approach.
Categories: hep-th math.QA