  We investigate Beurling-Fourier algebras, a weighted version of Fourier
algebras, on various Lie groups focusing on their spectral analysis. We will
introduce a refined general definition of weights on the dual of locally
compact groups and their associated Beurling-Fourier algebras. Constructions of
nontrivial weights will be presented focusing on the cases of representative
examples of Lie groups, namely $SU(n)$, the Heisenberg group $\mathbb{H}$, the
reduced Heisenberg group $\mathbb{H}_r$, the Euclidean motion group $E(2)$ and
its simply connected cover $\widetilde{E}(2)$. We will determine the spectrum
of Beurling-Fourier algebras on each of the aforementioned groups emphasizing
its connection to the complexification of underlying Lie groups. We also
demonstrate "polynomially growing" weights does not change the spectrum and
show the associated regularity of the resulting Beurling-Fourier algebras.
