2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/62625The paper contains a systematic theory of the one-dimensional Double Hecke algebra, including applications to the difference Fourier transform, Macdonald's polynomials, Gaussian sums at roots of unity, and Verlinde algebras. The main result is the classification of finite-dimensional representations for generic q and at roots of unity.Sections 8,9 are updatedQuantum AlgebraRepresentation Theorytext