Classification of quantum groups and Belavin-Drinfeld cohomologies

2013/03/17 by Kadets, Boris, Karolinsky, Eugene, Stolin, Alexander +1
#17B37 #17B81 #FOS: Mathematics #Quantum Algebra (math.QA)

paper · doi:10.48550/arxiv.1303.4046

Abstract

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

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