  An extensive analytical and numerical investigation has been carried out to
examine the role played by many-body effects on various
$\alpha$-$\mathcal{T}_3$ materials under an off-resonance optical dressing
field. Additionally, we explore its dependence on the hopping parameter
$\alpha$ as well as the electron-light coupling strength $\lambda_0$. The
obtained dressed states due to mutual interaction between Dirac electrons and
incident light are shown to demonstrate rather different electronic and optical
properties in comparison with those in the absence of incident light.
Specifically, various collective transport and optical properties of these
electron dressed states are discussed in detail and compared for both single-
and double layer $\alpha$-$\mathcal{T}_3$ lattices. All of these novel
properties are due to the presence of a middle flat band and the interband
transitions between it and an upper conduction band. Also, coupled plasmon
dispersions for interacting double layer $\alpha$-$\mathcal{T}_3$ lattices are
calculated, revealing a lower acoustic-like plasmon branch with tunable group
velocity determined by both the layer separation and Fermi energy of each
layer. Finally, a many-body theory is presented within the random-phase
approximation for calculating the optical absorbance of doped multi-layered
$\alpha$-$\mathcal{T}_3$ lattices in a linearly-polarized light field. We
anticipate that the discoveries reported here could impact the design of the
next-generation nano-optical and nano-plasmonic devices.
