Non-Abelian cubic vertices for higher-spin fields in AdSd

We use the Fradkin-Vasiliev procedure to construct the full set of non-Abelian cubic vertices for totally symmetric higher spin gauge fields in AdSd space. The number of such vertices is given by a certain tensor-product multiplicity. We discuss the one-to-one relation between our result and the list of non-Abelian gauge deformations in flat space obtained elsewhere via the cohomological approach. We comment about the uniqueness of Vasiliev’s simplest higher-spin algebra in relation with the (non)associativity properties of the gauge algebras that we classified. The gravitational interactions for (partially)-massless (mixed)-symmetry fields are also discussed. We also argue that those mixed-symmetry and/or partially-massless fields that are described by one-form connections within the frame-like approach can have non-Abelian interactions among themselves and again the number of non-Abelian vertices should be given by tensor product multiplicities.