  In this work, $\mathcal{PT}$-symmetric Hamiltonians defined on quantum $sl(2,
\mathbb R)$ algebras are presented. We study the spectrum of a family of
non-Hermitian Hamiltonians written in terms of the generators of the
non-standard $U_{z}(sl(2, \mathbb R))$ Hopf algebra deformation of $sl(2,
\mathbb R)$. By making use of a particular boson representation of the
generators of $U_{z}(sl(2, \mathbb R))$, both the co-product and the
commutation relations of the quantum algebra are shown to be invariant under
the $\mathcal{PT}$-transformation. In terms of these operators, we construct
several finite dimensional $\mathcal{PT}$-symmetry Hamiltonians, whose spectrum
is analytically obtained for any arbitrary dimension. In particular, we show
the appearance of Exceptional Points in the space of model parameters and we
discuss the behaviour of the spectrum both in the exact $\mathcal{PT}$-symmetry
and the broken $\mathcal{PT}$-symmetry dynamical phases. As an application, we
show that this non-standard quantum algebra can be used to define an effective
model Hamiltonian describing accurately the experimental spectra of
three-electron hybrid qubits based on asymmetric double quantum dots.
Remarkably enough, in this effective model, the deformation parameter $z$ has
to be identified with the detuning parameter of the system.
