An explicit product for the on-shell higher-spin algebra in $AdS_d$-space

Abstract: For $d>3$ we construct the product of the bosonic on-shell higher-spin algebra directly on the AdS-traceless representatives. We find the traceless projector in closed form, determine all linear relations among the trace-producing operators, and prove that every class modulo the $\mathfrak{sp}(2)$ ideal has a unique traceless representative. The resulting product is an explicit differential operator, which we test on the $\mathfrak{so}(d-1,2)$ generators. A toy model shows how the factorisation lifts from bilinear products to trilinear ones in minimal $A_\infty$-algebras.