  We present a numerical and analytical study of incompressible homogeneous
conducting fluids using a helical Fourier representation. We analytically study
both small- and large-scale dynamo properties, as well as the inverse cascade
of magnetic helicity, in the most general minimal subset of interacting
velocity and magnetic fields on a closed Fourier triad. We mainly focus on the
dependency of magnetic field growth as a function of the distribution of
kinetic and magnetic helicities among the three interacting wavenumbers. By
combining direct numerical simulations of the full magnetohydrodynamics
equations with the helical Fourier decomposition we numerically confirm that in
the kinematic dynamo regime the system develops a large-scale magnetic helicity
with opposite sign compared to the small-scale kinetic helicity, a sort of
triad-by-triad $\alpha$-effect in Fourier space. Concerning the small-scale
perturbations, we predict theoretically and confirm numerically that the
largest instability is achieved for the magnetic component with the same
helicity of the flow, in agreement with the Stretch-Twist-Fold mechanism. Vice
versa, in presence of a Lorentz feedback on the velocity, we find that the
inverse cascade of magnetic helicity is mostly local if magnetic and kinetic
helicities have opposite sign, while it is more nonlocal and more intense if
they have the same sign, as predicted by the analytical approach. Our
analytical and numerical results further demonstrate the potential of the
helical Fourier decomposition to elucidate the entangled dynamics of magnetic
and kinetic helicities both in fully developed turbulence and in laminar flows.
