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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
Article

Abstract

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.

Keywords

Quantum Group Cartan Subalgebra Twisted Cohomology Twisted Cocycle Trivial Cocycle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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