QUANTUM LIE-ALGEBRAS, THEIR EXISTENCE, UNIQUENESS AND Q-ANTISYMMETRY

Quantum Lie algebras are generalizations of Lie algebras which have th e quantum parameter h built into their structure. They have been defin ed concretely as certain submodules L-h(g) of the quantized enveloping algebras U-h(g). On them the quantum Lie product is given by the quan tum adjoint action. Here we define for any finite-dimensional simple c omplex Lie algebra g an abstract quantum Lie algebra g(h) independent of any concrete realization. Its h-dependent structure constants are g iven in terms of inverse quantum Clebsch-Gordan coefficients. We then show that all concrete quantum Lie algebras L-h(g) are isomorphic to a n abstract quantum Lie algebra g(h). In this way we prove two importan t properties of quantum Lie algebras: 1) all quantum Lie algebras L-h( g) associated to the same g are isomorphic, 2) the quantum Lie product of any Ch(B) is q-antisymmetric. We also describe a construction of L -h(g) which establishes their existence.