  We consider the Lie group $\mathbb{R}^D_\kappa$ generated by the Lie algebra
of $\kappa$-Minkowski space. Imposing the invariance of the metric under the
pull-back of diffeomorphisms induced by right translations in the group, we
show that a unique right invariant metric is associated with
$\mathbb{R}^D_\kappa$. This metric coincides with the metric of de Sitter
space-time. We analyze the structure of unitary representations of the group
$\mathbb{R}^D_\kappa$ relevant for the realization of the non-commutative
$\kappa$-Minkowski space by embedding into $(2D-1)$-dimensional Heisenberg
algebra. Using a suitable set of generalized coherent states, we select the
particular Hilbert space and realize the non-commutative $\kappa$-Minkowski
space as an algebra of the Hilbert-Schmidt operators. We define dequantization
map and fuzzy variant of the Laplace-Beltrami operator such that dequantization
map relates fuzzy eigenvectors with the eigenfunctions of the Laplace-Beltrami
operator on the half of de Sitter space-time.
