Classification of classical twists of the standard Lie bialgebra structure on a loop algebra
Artikel i vetenskaplig tidskrift, 2021

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. In this paper we classify all classical twists of the induced Lie bialgebra structures in terms of Belavin–Drinfeld quadruples up to a natural notion of equivalence. To obtain this classification we first show that the induced bialgebra structures are defined by certain solutions of the classical Yang–Baxter equation (CYBE) with two parameters. Then, using the algebro–geometric theory of CYBE, based on torsion free coherent sheaves, we reduce the problem to the well-known classification of trigonometric solutions given by Belavin and 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 CYBE.

Loop algebra

Kac–Moody algebra

Classical Yang–Baxter equation

Manin triple

Trigonometric r-matrices

Lie bialgebra


Raschid Abedin

Universität Paderborn

Stepan Maximov

Chalmers, Matematiska vetenskaper, Algebra och geometri

Göteborgs universitet

Journal of Geometry and Physics

0393-0440 (ISSN)

Vol. 164 104149


Algebra och logik


Matematisk analys



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