Belavin-Drinfeld cohomologies and introduction to classification of quantum groups

22nd International Conference on Integrable Systems and Quantum Symmetries

Stolin, Alexander^1 ; Pop, Iulia^1

Department of Mathematical Sciences, University of Goteborg, Göteborg

41296, Sweden^1

关键词: Bialgebras;    classical double;    Cohomology;    Lie Algebra;    Quantum groups;    Quasiclassical limits;    R matrices;    Skew-symmetric;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/563/1/012030/pdf
DOI  :  10.1088/1742-6596/563/1/012030

来源: IOP

【 摘 要 】

In the present article we discuss the classification of quantum groups whose quasiclassical limit is a given simple complex Lie algebra g. This problem reduces to the classification of all Lie bialgebra structures on g(K), where K C((h)). The associated classical double is of the form g(K)⊗KA, where A is one of the following: K[v], where v2=0, K ⊕ K or K[j], where j2h. The first case relates 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 from the Belavin-Drinfeld list [1]. 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.