Classification of classical twists of the standard Lie bialgebra structure on a loop algebra

Abstract: This licentiate thesis is based on the work "Classification of classical twists of the standard Lie bialgebra structure on a loop algebra" by R. Abedin and the author of this thesis. The standard Lie bialgebra structure on an affine Kac-Moody algebra induces a Lie bialgebra structure on the underlying loop algebra and its parabolic subalgebras. We study classical twists of the induced Lie bialgebra structures and obtain their full classification in terms of Belavin-Drinfeld quadruples up to a natural notion of equivalence. To obtain this classification we first show that the induced Lie bialgebra structures are determined by certain solutions of the classical Yang-Baxter equation (CYBE) with two parameters. Then, using the algebro-geometric theory of the CYBE, based on torsion free coherent sheaves, we reduce the problem to the well-known classification of trigonometric solutions given by A. Belavin and V. Drinfeld. The classification of twists in the case of parabolic subalgebras allows us to answer recently posed open questions regarding the so-called quasi-trigonometric solutions of the CYBE.