TY - JOUR
T1 - Classification of Quantum Groups and Belavin–Drinfeld Cohomologies
AU - Kadets, Boris
AU - Karolinsky, Eugene
AU - Pop, Iulia
AU - Stolin, Alexander
N1 - Publisher Copyright:
© 2016, Springer-Verlag Berlin Heidelberg.
PY - 2016/5/1
Y1 - 2016/5/1
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=84964397407&partnerID=8YFLogxK
U2 - 10.1007/s00220-016-2622-y
DO - 10.1007/s00220-016-2622-y
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AN - SCOPUS:84964397407
SN - 0010-3616
VL - 344
SP - 1
EP - 24
JO - Communications in Mathematical Physics
JF - Communications in Mathematical Physics
IS - 1
ER -