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

Abstract

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.

arXiv comment30 pages, 2 figures

References in corpus (4)

• On Some Lie Bialgebra Structures on Polynomial Algebras and their Quantization
• Classical dynamical r-matrices, Poisson homogeneous spaces, and Lagrangian subalgebras
• Algebraic geometry of Lie bialgebras defined by solutions of the classical Yang-Baxter equation
• Simple vector bundles on a nodal Weierstrass cubic and quasi-trigonometric solutions of CYBE

Cited by in corpus (2)

• Algebraic geometry of Lie bialgebras defined by solutions of the classical Yang-Baxter equation
• Topological Manin pairs and $(n,s)$-type series