An example of a Z-PBW algebra which is not a PBW algebra?

Question

Can anyone provide an example of a (quadratic) Z-PBW algebra which is not a (quadratic) PBW algebra? I am using the definition of Z-PBW algebra given in Polishchuk and Positselski's book Quadratic Algebras. I have tried verifying the assertion made about the algebra in the example at the bottom of page 97. However, to the best of my knowledge, that assertion appears to be false.

Answer 1

I have just rechecked the assertion in the Example at the bottom of page 97 and to the best of my understanding it is correct.

I can imagine one cause of a possible confusion, namely, the variables are not listed in the order of their increase in the sense of the ordering that makes them Z-PBW-generators. The order should be $x^{\sigma+1,\sigma}>y^{\sigma+1,\sigma}>z^{\sigma+1,\sigma}$, the order of monomials being the lexicographical order corresponding to this order of generators. It is presumed that the surviving monomials in the PBW-basis are those that cannot be expressed as linear combinations of any smaller ones (modulo the relations).

Specifically, the quadratic monomials that do not survive (do not belong to the set $S^{\sigma+1,\sigma-1}$) are $x^{\sigma+1,\sigma}y^{\sigma,\sigma-1}$ and $x^{\sigma+1,\sigma}z^{\sigma,\sigma-1}$. Given their form, it is immediately clear that the PBW condition is satisfied.

UPDATE. Well, another source of a possible confusion is a misprint in our formulas. It should be $b_{\sigma+1}=-1/c_\sigma$ and $c_{\sigma+1}=-1/(ab_\sigma)$ and not the other way around. To put it simply, the constants $b_\sigma$ and $c_\sigma$ are chosen in such a way that the terms $x^{\sigma+1,\sigma}x^{\sigma,\sigma-1}$ cancel out in each relation.

Answer 2

Thank you very much Dr. Positselski. I now agree that the example is correct.

Perhaps you could shed some light on the following (vague) questions.

(1) Are there any hints or reasons to suspect a given quadratic algebra is Z-PBW? In your example one can get rid of "inclusion ambiguity" (the common $x^2$ terms of the relations) using the Z-PBW idea.

(2) Do you know any examples in the literature of quadratic algebras whose Koszulity has been proved by verifying Z-PBW (without PBW)?

I am trying to prove Koszulity of some algebras which do not appear to be PBW. However I have found some monomial orderings for which there are only a few "overlap ambiguities". I'm using the terminology in Bergman's paper on the Diamond Lemma.