Abstract:
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 of the quantized enveloping algebras . On them the quantum Lie product is given by the quantum adjoint action.
Here we define for any finite-dimensional simple complex Lie algebra an abstract quantum Lie algebra 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 are isomorphic to an abstract quantum Lie algebra .
In this way we prove two important properties of quantum Lie algebras: 1) all quantum Lie algebras associated to the same are isomorphic, 2) the quantum Lie product of any is q-antisymmetric. We also describe a construction of which establishes their existence.
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Received: 23 May 1996 / Accepted: 17 October 1996
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Delius, G., Gould, M. Quantum Lie Algebras, Their Existence, Uniqueness and q-Antisymmetry . Comm Math Phys 185, 709–722 (1997). https://doi.org/10.1007/s002200050107
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DOI: https://doi.org/10.1007/s002200050107