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?
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4$\begingroup$ What is the paper? $\endgroup$– LSpiceOct 28, 2022 at 17:57
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2$\begingroup$ Have you tried Googling "PBW algebra" with the quotation marks (or maybe I should say ""PBW algebra"")? It gives me several references, although one of them seems to suggest that there may be more than one nonequivalent definitions. $\endgroup$– Jeremy RickardOct 28, 2022 at 18:32
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1$\begingroup$ @OP, although your question is clearly stated (aside from omitting the name of the paper) and makes the background clear, you've just got your second answer from someone who thinks you're asking about the PBW theorem. Though it's through no fault of your own, you might find it helpful to include a big bold disclaimer up front that you do know the PBW theorem and that you are only looking for a reference to whether your understanding of its generalisation to abstract algebras is correct. $\endgroup$– LSpiceOct 28, 2022 at 19:59
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2$\begingroup$ The are various definitions of what that is, really, not all equivalent. What is the paper? $\endgroup$– Mariano Suárez-ÁlvarezOct 28, 2022 at 23:47
1 Answer
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].
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$\begingroup$ I would add that there are some specializations and generalizations of this notion that appeared since 1970, and that the survey of Anne Shepler and Sarah Witherspoon arxiv.org/abs/1404.6497 is a good place to look at to account for that. $\endgroup$ Oct 30, 2022 at 16:31