Classification of Lie bialgebras over current algebras (Q607474)

This paper presents a classification of Lie bialgebras structures on the Lie algebras of power series \({\mathfrak g}[[x]]\), where \({\mathfrak g}\) is a finite dimensional, complex, simple Lie algebra. This is achieved through a description of the possible corresponding Drinfeld doubles. Since a Lie bialgebra structure \(\delta :{\mathfrak g}[x]\to {\mathfrak g}[x]\otimes{\mathfrak g}[y]\) can always be extended to some \(\overline{\delta}:{\mathfrak g}[[x]]\to {\mathfrak g}\otimes{\mathfrak g}[[x,y]]\) (Lemma 4.1) the above result can be applied for the classification of doubles and Lie bialgebras structures on the Lie algebras of type \({\mathfrak g}[x]\). Two different Lie bialgebra structures on the same Lie algebra yielding the same Drinfeld double are obtained from one another by means of a classical twist. A classification of classical twists for all Lie algebra structures on \({\mathfrak g}[x]\) is provided. Some remarks on quantization of the different Lie bialgebras structures are also given.