Quantum Lie algebras , their existence , uniqueness and q - antisymmetry

Quantum Lie algebras are generalizations of Lie algebras which have the quantum parameter h built into their structure. They have been defined concretely as certain submodules Lh(g) of the quantized enveloping algebras Uh(g). On them the quantum Lie bracket is given by the quantum adjoint action. Here we define for any finite-dimensional simple complex Lie algebra g an abstract quantum Lie algebra gh independent of any concrete realization. Its h-dependent structure constants are given in terms of inverse quantum Clebsch-Gordan coefficients. We then show that all concrete quantum Lie algebras Lh(g) are isomorphic to an abstract quantum Lie algebra gh. In this way we prove two important properties of quantum Lie algebras: 1) all quantum Lie algebras Lh(g) associated to the same g are isomorphic, 2) the quantum Lie bracket of any Lh(g) is qantisymmetric. We also describe a construction of Lh(g) which establishes their existence.