I am reading a paper where they refer to a certain algebra as a **PBW algebra**. What does this mean exactly? I would infer from the $U(\frak{g})$ setting that this means the existence of an ordered generating set $\{x\_1, \dots, x\_n\}$ such that
$$
x\_1^{d\_1} \cdots x\_n^{d\_n}, ~~~~~~ d\_k \in \mathbb{N}\_0.
$$
forms a basis for the algebra. Can somebody point me to a reference?



What is a PBW algebra? (I.e., an algebra generalising properties of $U(\frak{g})$)
This concept, and the name Poincare Birkhoff Witt algebra, is due to Stewart Priddy in his paper [Koszul resolutions, Trans. Amer. Math. Soc. 152 (1970), 39–60] that also first defined Koszul algebras. See section 5 of his paper for the PBW criterion, which is a bit more flexible than what you wrote. (The basis might not need all of those words.)


Added later: The reference to the published version of the nice survey paper that Vladimir Dotsenko mentions is [Shepler, Anne V.; Witherspoon, Sarah Poincaré-Birkhoff-Witt theorems. Commutative algebra and noncommutative algebraic geometry. Vol. I, 259–290, Math. Sci. Res. Inst. Publ., 67, Cambridge Univ. Press, New York, 2015].


