  We discuss a new class of strong homotopy algebras constructed via inner
deformations. Such deformations have a number of remarkable properties. In the
simplest case, every one-parameter family of associative algebras leads to an
$L_\infty$-algebra that can be used to construct a classical integrable model.
Another application of this class of $L_\infty$-algebras is related with the
three-dimensional bosonization duality in Chern--Simons vector models, where it
implements the idea of the slightly-broken higher spin symmetry. One large
class of associative algebras originates from Deformation Quantization of
Poisson Manifolds. Applications to the $3d$-bosonization duality require,
however, an extension to deformation quantization of Poisson Orbifolds, which
is an open problem. The $3d$-bosonization duality can be proven by showing that
there is a unique class of invariants of the $L_\infty$-algebra that can serve
as correlation functions.
