2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/70881This paper is based on the introduction to the monograph ``Double affine Hecke algebras'' to be published by Cambridge University Press. The connections with Knizhnik-Zamolodchikov equations, Kac-Moody algebras, tau-function, harmonic analysis on symmetric spaces, and special functions are discussed. The rank one case is considered in detail including the classification of Verlinde algebras and their deformations, Gauss-Selberg integrals and Gaussian sums, a topological interpretation of DAHA, a relation of the rational DAHA to sl(2), and applications to the diagonal coinvariants. The last three sections are devoted to relations of the general DAHAs to the p-adic affine Hecke algebras, trigonometric and rational DAHAs, and applications to the Harish-Chandra theory. The purpose of this introduction is a demonstration that DAHA can be considered as a natural formalization of the concept of the Fourier transform in mathematics and physics.LaTeX, 93 pgs, 7 figures, a significantly extended variantQuantum AlgebraMathematical PhysicsCombinatoricsGeometric TopologyRepresentation Theory22Exx,33Cxx,33Dxx,81Rxx,05E05,11T24,14H52,14J25,14M12,14M15,16S90, 20B30,20F34,20F36,55R80text