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Aug 23, 2017 at 17:18 comment added Konstantinos Kanakoglou you are right. The quantum inverse scattering method, its application(s) in quantum integrable systems (among which the Heisenberg quantum spin chain has a special historical importance) and the consequent developments in $q$-deformations of Lie algebras, quantum groups, etc have gone hand-by-hand with the development and the applications of $q$-mathematics. These examples certainly deserve to be mentioned on their own.
Aug 22, 2017 at 14:55 comment added Jules Lamers I should probably also mention applications of $q$-mathematics in integrable quantum many-body models (Calogero--Sutherland, Ruijsenaars--Schneiders), where Hecke algebras, Macdonald polynomials, etc play an important role in the model's analysis.
Aug 22, 2017 at 13:14 history answered Jules Lamers CC BY-SA 3.0