Communications in Mathematical Physics

, Volume 344, Issue 1, pp 1–24 | Cite as

Classification of Quantum Groups and Belavin–Drinfeld Cohomologies

  • Boris Kadets
  • Eugene Karolinsky
  • Iulia Pop
  • Alexander Stolin


In the present article we discuss the classification of quantum groups whose quasi-classical limit is a given simple complex Lie algebra \({\mathfrak{g}}\). This problem is reduced to the classification of all Lie bialgebra structures on \({\mathfrak{g}(\mathbb{K})}\), where \({\mathbb{K}=\mathbb{C}((\hbar))}\). The associated classical double is of the form \({\mathfrak{g}(\mathbb{K})\otimes_{\mathbb{K}} A}\), where A is one of the following: \({\mathbb{K}[\varepsilon]}\), where \({\varepsilon^{2}=0}\), \({\mathbb{K}\oplus\mathbb{K}}\) or \({\mathbb{K}[j]}\), where \({j^{2}=\hbar}\). The first case is related to quasi-Frobenius Lie algebras. In the second and third cases we introduce a theory of Belavin–Drinfeld cohomology associated to any non-skewsymmetric r-matrix on the Belavin–Drinfeld list (Belavin and Drinfeld in Soviet Sci Rev Sect C: Math Phys Rev 4:93–165, 1984). We prove a one-to-one correspondence between gauge equivalence classes of Lie bialgebra structures on \({\mathfrak{g}(\mathbb{K})}\) and cohomology classes (in case II) and twisted cohomology classes (in case III) associated to any non-skewsymmetric r-matrix.


Quantum Group Cartan Subalgebra Twisted Cohomology Twisted Cocycle Trivial Cocycle 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Boris Kadets
    • 1
  • Eugene Karolinsky
    • 1
  • Iulia Pop
    • 2
  • Alexander Stolin
    • 2
  1. 1.Department of Mechanics and MathematicsKharkov National UniversityKharkovUkraine
  2. 2.Department of MathematicsGothenburg UniversityGothenburgSweden

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