  Among non-Hermitian systems, pseudo-Hermitian phases represent a special
class of physical models characterized by real energy spectra and by the
absence of non-Hermitian skin effects. Here, we show that several
pseudo-Hermitian phases in two and three dimensions can be built by employing
$q$-deformed matrices, which are related to the representation of deformed
algebras. Through this algebraic approach we present and study the
pseudo-Hermitian version of well known Hermitian topological phases, raging
from two-dimensional Chern insulators and time-reversal-invariant topological
insulators to three-dimensional Weyl semimetals and chiral topological
insulators. We analyze their topological bulk states through non-Hermitian
generalizations of Abelian and non-Abelian tensor Berry connections and quantum
metric. Although our pseudo-Hermitian models and their Hermitian counterparts
share the same topological invariants, their band geometries are different. We
indeed show that some of our pseudo-Hermitian phases naturally support
nearly-flat topological bands, opening the route to the study of
pseudo-Hermitian strongly-interacting systems. Finally, we provide an
experimental protocol to realize our models and measure the full non-Hermitian
quantum geometric tensor in synthetic matter.
