Dear Colleagues,

Symmetry is one of the most essential ingredients of integrability. By the very definition of integrability, integrable systems have Abelian symmetries as a consequence of the existence of conservation laws. Moreover, since the late 1970s, diverse types of non-Abelian symmetries have been discovered. Those non-Abelian symmetries are accompanied by particular algebraic structures. For instance, integrable many-body problems of the Calogero and Toda type are characterized by simple Lie algebras and associated Weyl groups. This is also the case for the 1+1 dimensional Drinfeld-Sokolov hierarchies. The 1+2 dimensional KP and Toda hierarchies and 1+1 dimensional reductions thereof are governed by infinite dimensional general linear, orthogonal and affine algebras. In quantum integrable systems, these Lie algebraic structures are deformed to “quantum algebras”, i.e., Yangians, quantum affine algebras, and their elliptic analogs. In a different context, quantization of the Drinfeld-Sokolov hierarchies is linked with symmetry algebras of conformal field theories. This Special Issue is devoted to more recent progress of these subjects and related issues.

Prof. Dr. Kanehisa Takasaki
Guest Editor

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• classical and quantum integrable systems
• solvable statistical models
• conformal field theories
• non-Abelian symmetries
• discrete symmetries
• simple and affine Lie algebras
• vertex operators
• Yangian, quantum affine and elliptic algebras
• Virasoro and W algebras
• Weyl groups, Coxeter groups and Hecke algebras