Rankin-Cohen algebras and $\mathfrak{sl}_2$-algebras

I will talk mainly about two algebraic structures and the relations between them.  The first is that of a {\it Rankin-Cohen algebra}, which is a graded vector space with infinitely many bilinear 
operations satisfying the same algebraic identities as those satisfied by the Rankin-Cohen brackets in the theory of modular forms.  The second is that of {\it $\mathfrak{sl}_2$-algebra}, 
meaning an algebra on which the 3-dimensional Lie algebra $\mathfrak{sl}_2$ acts by derivations.  It turns out that there is a natural isomorphism (actually two, depending on the precise conditions that we impose) between the category of Rankin-Cohen algebras and the category of commutative and asso\-ciative \hbox{$\mathfrak{sl}_2$-algebras,} both motivated by ideas coming from elliptic modular forms and Siegel modular forms.  There is also a non-commutative version based on work that I did many years ago with Yuri Manin and which led to the construction of infinitely many associative multiplications on algebras of modular forms.  Finally, although I will only touch on this very briefly, there is an analogous story relating conformal algebras (a variant of the better known notions of vertex algebras and vertex opeator algebras from conformal field theory) to $\mathfrak{sl}_2$-algebras, but this time to $\mathfrak{sl}_2$-Lie algebras rather than to associative ones.