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Poincare-Birkhoff-Witt (PBW) Theorems have attracted significant attentionsince 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 forother cocommutative Hopf algebras $H$. In this paper we show that such PBWtheorems do not require the full Hopf algebra structure, by working in the moregeneral setting of a "cocommutative algebra", which involves a coproduct butnot a counit or antipode. Special cases include infinitesimal Hecke algebras,as well as symplectic reflection algebras, rational Cherednik algebras, andmore generally, Drinfeld orbifold algebras. In this generality we identifyprecise conditions that are equivalent to the PBW property, including aYetter-Drinfeld type compatibility condition and a Jacobi identity. We alsocharacterize the graded deformations that possess the PBW property. In turn,the PBW property helps identify an analogue of symplectic reflections ingeneral cocommutative bialgebras.  Next, we introduce a family of cocommutative algebras outside thetraditionally studied settings: generalized nil-Coxeter algebras. These arenecessarily not Hopf algebras, in fact, not even (weak) bialgebras. For thecorresponding family of deformed smash product algebras, we compute the centeras well as abelianization, and classify all simple modules.

Generalized nil-Coxeter algebras, cocommutative algebras, and the PBW property