  The Hofstadter model allows to describe and understand several phenomena in
condensed matter such as the quantum Hall effect, Anderson localization, charge
pumping, and flat-bands in quasiperiodic structures, and is a rare example of
fractality in the quantum world. An apparently unrelated system, the
relativistic Toda lattice, has been extensively studied in the context of
complex nonlinear dynamics, and more recently for its connection to
supersymmetric Yang-Mills theories and topological string theories on
Calabi-Yau manifolds in high-energy physics. Here we discuss a recently
discovered spectral relationship between the Hofstadter model and the
relativistic Toda lattice which has been later conjectured to be related to the
Langlands duality of quantum groups. Moreover, by employing similarity
transformations compatible with the quantum group structure, we establish a
formula parametrizing the energy spectrum of the Hofstadter model in terms of
elementary symmetric polynomials and Chebyshev polynomials. The main tools used
are the spectral duality of tridiagonal matrices and the representation theory
of the elementary quantum group.
