Read/Search this Article
We review the differential calculus on quantum groups following the approach used by Woronowicz. It leads us to introduce two notions of quantum Lie algebras which we refer to as braided or quasi-triangular quantum Lie algebras. They are both different generalizations of the algebras defined by Gurevitch. They are characterized by quadratic relations and braided commutators. The quasi-triangular quantum Lie algebras can be defined as exchange algebras. To any quantum Lie algebra we associate a quantum group with a differential calculus on it such that the algebra of the quantum Lie derivatives is the quantum Lie algebra. We mention two possible `physical' applications: gauge theories on quantum groups and non-local currents in two-dimensional quantum field theories.