  We investigate the deformations and rigidity of boundary Heisenberg-like
algebras. In particular, we focus on the Heisenberg and
$\text{Heisenberg}\oplus\mathfrak{witt}$ algebras which arise as symmetry
algebras in three-dimensional gravity theories. As a result of the deformation
procedure we find a large class of algebras. While some of these algebras are
new, some of them have already been obtained as asymptotic and boundary
symmetry algebras, supporting the idea that symmetry algebras associated to
diverse boundary conditions and spacetime loci are algebraically interconnected
through deformation of algebras. The deformation/contraction relationships
between the new algebras are investigated. In addition, it is also shown that
the deformation procedure reaches new algebras inaccessible to the Sugawara
construction. As a byproduct of our analysis, we obtain that
$\text{Heisenberg}\oplus\mathfrak{witt}$ and the asymptotic symmetry algebra
Weyl-$\mathfrak{bms}_3$ are not connected via single deformation but in a more
subtle way.
