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Algebraic geometry of the classical Yang-Baxter equation and its generalizations / von Raschid Abedin. Paderborn, 2022
Inhalt
Introduction
Content and results
Conventions in notation and terminology
Some facts about sheaves of algebras
Basic definitions and properties
Local triviality of sheaves of algebras
Lattices in current algebras and sheaves of algebras on projective curves
Formal generalized r-matrices
Basic definitions and properties
Lie subalgebras of g((z)) complementary to g[[z]]
Formal r-matrices
Algebro-geometric properties of formal generalized r-matrices
Geometrization of formal generalized r-matrices.
Properties of geometric data associated to formal generalized r-matrices
Global aspects of formal generalized r-matrices
Analytic generalized r-matrices
Twisted loop algebras
Kac-Moody algebras
Twisted Loop algebras and affine Kac-Moody algebras
Lie bialgebras
Basic definitions and properties
Manin triples
Twisting Lie bialgebra structures
Examples of Lie bialgebras
The Belavin-Drinfeld trichotomy
Sheaves of algebras on algebraic groups of dimension one
The trichotomy theorem
Elliptic r-matrices
Acyclic weakly g-locally free sheaves of algebras on elliptic curves.
Classification of elliptic r-matrices.
Trigonometric r-matrices
Explicit geometrization
Classification
Equivalences
Restrictions of the standard structure
Rational r-matrices
Explicit geometrization
Structure theory
Bibliography
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