Towards a Cartan Classification of Quantum Algebras

Abstract. We present here the first attempts for a classification of quantum algebras in the spirit of the Cartan classification of semisimple Lie algebras. We introduce a ‘perturbative ’ quantization procedure of Lie bialgebras where ana-lyticity in the deformation parameter plays an essential role. Our classification is up to Hopf algebra isomorphisms preserving a form-invariant coproduct. As an example we classify the three dimensional quantum algebras. Also we extend this method to Drinfel’d doubles in order to work in higher dimensions. PACS number: 02.20.Sv