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Bulgarian Journal of Physics
Print: 1310-0157 Online: 1314-2666
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RECOGNIZED BY THE EUROPEAN PHYSICAL SOCIETY 
Bulg. J. Phys. vol.33 no.s2 (2006), pp. 85-96
Towards a Cartan Classification of Quantum Algebras
M.A. del Olmo1, A. Ballesteros2, E. Celeghini3
1Departamento de Física Teórica, Universidad de Valladolid, E-47011, Valladolid, Spain
2Departamento de Física, Universidad de Burgos, E-09006, Burgos, Spain
3Dipartimento di Fisica, Universitá di Firenze and INFN–Sezione di Firenze I50019 Sesto Fiorentino, Firenze, Italy
go back1Departamento de Física Teórica, Universidad de Valladolid, E-47011, Valladolid, Spain
2Departamento de Física, Universidad de Burgos, E-09006, Burgos, Spain
3Dipartimento di Fisica, Universitá di Firenze and INFN–Sezione di Firenze I50019 Sesto Fiorentino, Firenze, Italy
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 analyticity 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.