  We present an approach to interacting quantum many-body systems based on the
notion of quantum groups, also known as $q$-deformed Lie algebras. In
particular, we show that if the symmetry of a free quantum particle corresponds
to a Lie group $G$, in the presence of a many-body environment this particle
can be described by a deformed group, $G_q$. Crucially, the single deformation
parameter, $q$, contains all the information about the many-particle
interactions in the system. We exemplify our approach by considering a quantum
rotor interacting with a bath of bosons, and demonstrate that extracting the
value of $q$ from closed-form solutions in the perturbative regime allows one
to predict the behavior of the system for arbitrary values of the impurity-bath
coupling strength, in good agreement with non-perturbative calculations.
Furthermore, the value of the deformation parameter allows to predict at which
coupling strengths rotor-bath interactions result in a formation of a stable
quasiparticle. The approach based on quantum groups does not only allow for a
drastic simplification of impurity problems, but also provides valuable
insights into hidden symmetries of interacting many-particle systems.
