2008/03/10 by A. Ballesteros, Ángel Ballesteros, E. Celeghini +1
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebra over a field #Algebraic structures and combinatorial models #Basis (linear algebra) #Bialgebra #Geometry #Hopf algebra #Lie algebra #Mathematics #Non-associative algebra #Nonlinear Waves and Solitons #Physics #Pure mathematics #Quantum #Quantum group #Quantum mechanics #Simple (philosophy) #Universal enveloping algebra #math.GR #math.QA #msc:17B37 #msc:81R40 #msc:81R50
paper · pdf · doi:10.1088/1742-6596/128/1/012049
16 pages, Proceedings of the 5th International Symposium on Quantum Theory and Symmetries QTS5 (July 22-28, 2007, Valladolid (Spain))
arxiv created 2008/03/10 · openalex publication_date 2008/08/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Applications of algebras in physics are related to the connection of measurable observables to relevant elements of the algebras, usually the generators. However, in the determination of the generators in Lie algebras there is place for some arbitrary conventions. The situation is much more involved in the context of quantum algebras, where inside the quantum universal enveloping algebra, we have not enough primitive elements that allow for a privileged set of generators and all basic sets are equivalent. In this paper we discuss how the Drinfeld double structure underlying every simple Lie bialgebra characterizes uniquely a particular basis without any freedom, completing the Cartan program on simple algebras. By means of a perturbative construction, a distinguished deformed basis (we call it the analytical basis) is obtained for every quantum group as the analytical prolongation of the above defined Lie basis of the corresponding Lie bialgebra. It turns out that the whole construction is unique, so to each quantum universal enveloping algebra is associated one and only one bialgebra. In this way the problem of the classification of quantum algebras is moved to the classification of bialgebras. In order to make this procedure more clear, we discuss in detail the simple cases of su (2) and su q (2).