Classification of Lie bialgebras over current algebras

Abstract

In this paper, we give a classification of Lie bialgebra structures on Lie algebras of type g[[x]] and g [x], where g is a simple complex finite dimensional Lie algebra.

On the General Classification of Lie Bialgebra Structures over Polynomials

“…One can easily check that¯ ⊕ ∩ u = and thus we have a correspondence between W and Lagrangian subalgebras W of ⊕ transversal to . Statement (ii) was proved in [3]. This ends the proof.…”

“…For any Lie cobracket on u , denote by¯ the induced Lie cobracket on u and by D¯ u the corresponding classical double. In [3], the following result was proved. In [5], we treated this infinite-dimensional problem by first showing that any such W can be embedded into a special algebra, denoted by , indexed by vertices of the extended Dynkin diagram of .…”

“…According to [3], a u cannot be arbitrarily chosen, there are certain restrictions to be imposed. In case I, 1/a u is a polynomial of degree at most 2, in case II, is at most 1 and in case III, its degree is 0.…”

“…The aim of the present article is to complete the classification of Lie bialgebra structures on u that was initiated in [3,5]. According to the procedure given in [3,5], the classification of Lie bialgebra structures is done in several steps.…”

“…The aim of the present article is to complete the classification of Lie bialgebra structures on u that was initiated in [3,5]. According to the procedure given in [3,5], the classification of Lie bialgebra structures is done in several steps. The problem can be first reduced to the classification of bounded Lagrangian subalgebras W of one of the following Lie algebras: u , u ⊕ or u ⊕ + , where 2 = 0, and which are transversal to the algebra of Taylor series u .…”