TITLE: Which finite posets are Koszul self-dual?
QUESTION [9 upvotes]: Let $P$ be a finite connected poset with incidence algebra $A_P$.
For the definition and results on Koszul algebras for incidence algebras, see for example here

Question: Which posets have the property that $A_P$ is a Koszul algebra such that the Koszul dual of $A_P$ is isomorphic to $A_P$?

It seems this property is extremely rare and only the finest posets are in this class such as Boolean lattices (and no other distributive lattices ?!) or the strong Bruhat order for the symmetric group.

REPLY [10 votes]: Elaborating on Richard's suggestion, I think that it is necessary (but don't know about sufficiency) for the poset $P$ to be Gorenstein* over the field $k$ used in defining the incidence algebra $A_P$. To see this, note that Koszulity of $A_P$ is equivalent to Cohen-Macaulay-ness of $P$ over $k$, by the results of Polo [23], Woodcock [29] (reference numbers from the paper that you quoted). Then Eulerian-ness is required by the numerology relating Hilbert series of a Koszul algebra to that of its quadratic dual, as in the reference [1, Lemma 2.11.1] by Beilinson, Ginzburg, and Soergel.  And as Richard said, the Gorenstein* property is defined as being both Cohen-Macaulay and Eulerian.