The involutive system of higher-spin equations

We revisit the problem of consistent free propagation higher-spin fields in nontrivial backgrounds, focusing on symmetric tensor(-spinor)s. The Fierz-Pauli equations for massive flat space form an involutive system, whose algebraic consistency owes to certain gauge identities. zero mass limit former leads directly massless transverse-traceless gauge, where both field and parameter have their respective systems In it is preservation these identities symmetries that ensures correct number propagating degrees freedom. With this approach we find sets bosons fermions gravitational/electromagnetic backgrounds. also present system partially fields, give explicit transformations. consider Lie superalgebra operators tensor(-spinor)s space, show AdS algebra closes nonlinearly requires a central extension.