Covariant Lagrangian Cubic Interaction Vertices For Irreducible Higher Spin Fields in Minkowski Backgrounds

We review the application of BRST and BRST-BV approaches to construct the generic off-shell local Lorentz covariant cubic interaction vertices for irreducible massless and massive higher integer spin fields (as the candidates for massive particles in the Dark Matter problem) on $d$-dimensional Minkowski spaces. It is shown that equivalence among two Lagrangian dynamics for the same cubically interacting fields with given masses and spins obtained by means of the approach with complete BRST, $Q$, operator and of one with incomplete BRST, $Q_c$, operator in presence of consistent off-shell holonomic (traceless) constraints can be uplifted from the equivalence of the Lagrangians for free higher spin fields. We found that to get non-contradictory Lagrangians for irreducible interacting higher-spin fields within approach with $Q_c$ operator, together with off-shell algebraic constraints in addition to necessary condition of superconmmuting of $Q_c$ with appropriate holonomic constraints on the field and gauge parameter vectors, these constraints should form Abelian superalgebra both with BRST operator above and with operators of cubic vertices.