  If A is a cocommutative algebra with coproduct, then so is the smash product
algebra of a symmetric algebra Sym(V) with A, where V is an A-module. Such
smash product algebras, with A a group ring or a Lie algebra, have families of
deformations that have been studied widely in the literature; examples include
symplectic reflection algebras and infinitesimal Hecke algebras. We introduce a
family of deformations of these smash product algebras for general A, and
characterize the PBW property. We then characterize the Jacobi identity for
"grouplike" algebras (that include group rings and the nilCoxeter algebra), and
precisely identify the PBW deformations in the example where A is the
nilCoxeter algebra. We end with the more prominent case - where A is a Hopf
algebra. We show the equivalence of several versions of the "deformed"
relations in the smash product, and identify the PBW deformations which are
Hopf algebras as well.
