Multiparameter Quantum Group and Quantum Minkowski Space-Time

Abstract—We construct representations of the quantum algebras ${{U}_{{q,{\mathbf{q}}}}}(gl(n))$ and ${{U}_{{q,{\mathbf{q}}}}}(sl(n))$ which depend on $n(n - {{1)} \mathord{\left/ {\vphantom {{1)} 2}} \right. \kern-0em} 2} + 1$ deformation parameters $q,{{q}_{{ij}}}$ ($1 \leqslant i < j \leqslant n$) which is the maximal possible number in the case of $GL(n).$ The representations act on the space of formal power series of $n(n - {{1)} \mathord{\left/ {\vphantom {{1)} 2}} \right. \kern-0em} 2}$ non-commuting variables which generate quantum flag manifolds of $G{{L}_{{q{\mathbf{q}}}}}(n),$$S{{L}_{{q{\mathbf{q}}}}}(n).$ For $n = 4$ we consider in detail the multiparameter quantum Minkowski space-time.