Classification of Quantum Groups and Belavin-Drinfeld Cohomologies
Journal article, 2016

In the present article we discuss the classification of quantum groups whose quasi-classical limit is a given simple complex Lie algebra g. This problem is reduced to the classification of all Lie bialgebra structures on g(K) , where K= C((ħ)). The associated classical double is of the form g(K) ⊗ KA, where A is one of the following: K[ ε] , where ε2= 0 , K⊕ K or K[ j] , where j2= ħ. 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 g(K) and cohomology classes (in case II) and twisted cohomology classes (in case III) associated to any non-skewsymmetric r-matrix.


B. Kadets

E. Karolinsky

Iulia Pop

Chalmers, Mathematical Sciences

University of Gothenburg

Alexander Stolin

Chalmers, Mathematical Sciences, Mathematics

University of Gothenburg

Communications in Mathematical Physics

0010-3616 (ISSN) 1432-0916 (eISSN)

Vol. 344 1 1-24

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