  Poincare-Birkhoff-Witt (PBW) Theorems have attracted significant attention
since the work of Drinfeld (1986), Lusztig (1989), and Etingof-Ginzburg (2002)
on deformations of skew group algebras $H \ltimes {\rm Sym}(V)$, as well as for
other cocommutative Hopf algebras $H$. In this paper we show that such PBW
theorems do not require the full Hopf algebra structure, by working in the more
general setting of a "cocommutative algebra", which involves a coproduct but
not a counit or antipode. Special cases include infinitesimal Hecke algebras,
as well as symplectic reflection algebras, rational Cherednik algebras, and
more generally, Drinfeld orbifold algebras. In this generality we identify
precise conditions that are equivalent to the PBW property, including a
Yetter-Drinfeld type compatibility condition and a Jacobi identity. We also
characterize the graded deformations that possess the PBW property. In turn,
the PBW property helps identify an analogue of symplectic reflections in
general cocommutative bialgebras.
  Next, we introduce a family of cocommutative algebras outside the
traditionally studied settings: generalized nil-Coxeter algebras. These are
necessarily not Hopf algebras, in fact, not even (weak) bialgebras. For the
corresponding family of deformed smash product algebras, we compute the center
as well as abelianization, and classify all simple modules.
