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We discuss a new class of strong homotopy algebras constructed via innerdeformations. Such deformations have a number of remarkable properties. In thesimplest 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 thethree-dimensional bosonization duality in Chern--Simons vector models, where itimplements the idea of the slightly-broken higher spin symmetry. One largeclass of associative algebras originates from Deformation Quantization ofPoisson Manifolds. Applications to the $3d$-bosonization duality require,however, an extension to deformation quantization of Poisson Orbifolds, whichis an open problem. The $3d$-bosonization duality can be proven by showing thatthere is a unique class of invariants of the $L_\infty$-algebra that can serveas correlation functions.

Integrable Models From Non-Commutative Geometry With Applications to 3D Dualities