Let $A$ be a quantum $\mathbb{P}^n$ defined by $$ A=\mathbb{C}\langle x_1,x_2,\dots,x_{n+1}\rangle/(x_ix_j-r_{ij}x_jx_i)_{1\le i < j\le n+1}. $$ I would like to know the set $X$ of isomorphism classes of point modules for $A$. Here a point module is a cyclic graded right $A$-module $M$ such that each graded piece of $M$ is one-dimensional.

The set $Y$ of isomorphism classes of of point modules for the quantum $\mathbb{P}^2$ $$ \mathbb{C}\langle x_1,x_2,x_3\rangle/(x_ix_j-r_{ij}x_jx_i)_{1\le i < j\le 3} $$ is known to be $\mathbb{P}^2$ or a union of three lines in $\mathbb{P}^2$.

It seems known that $X$ is projective for $n=4$, but does anyone know explicit description of $X$? I tried to compute $X$ in a similar manner as the quantum $\mathbb{P}^2$ case, but it seems quite complicated. I would also appreciate it if someone give me a good description for higher dimensional $n$ case (especially $n=3,4$).

Thank you very much.

  • $\begingroup$ A possible way to do this is to view this ring as a Zhang twist of a polynomial ring on $n$ variables (I think this is possible). The paper where these first came up is "Twisted graded algebras and equivalences of graded categories". If they are Zhang twists of each other then they have isomorphic graded module categories (Thm. 1.2) and I think it follows that their point variety is $\mathbb{P}^n$ under some conditions on the scalars. But there are lots of ifs, buts and maybes here.... $\endgroup$ – Andrew Davies Oct 11 '12 at 10:47
  • $\begingroup$ Is it true that commutative analogs of these modules are just ideal of functions vanishing at some point ? What about P^1 ? How to describe the modules over P^2 ? are there explicit examples ? $\endgroup$ – Alexander Chervov Oct 11 '12 at 11:50
  • $\begingroup$ In "Noncommutative Instantons and Twistor Transform" by Kuznetsov Orlov Kapustin arxiv.org/abs/hep-th/0002193 in section 5 page 21 they describe the some modules over P^3_q ... I am not sure the answer is there, but may be ... $\endgroup$ – Alexander Chervov Oct 11 '12 at 12:02
  • $\begingroup$ Have you looked at Rogalski's notes? He mentions that in general X will live in the product of projective spaces. He also talks about how consider them for a finitely presented algebra with homogeneous ideal. $\endgroup$ – B. Bischof Oct 11 '12 at 12:44
  • $\begingroup$ Thank you for the comments. I will take a look at the papers you suggested. $\endgroup$ – user2013 Oct 12 '12 at 4:38

$X$ is either isomorphic to $\mathbb{P}^n$ or is the union of some faces of the fundamental $n$-simplex (on the points $v_i = [\delta_{1i} : ... : \delta_{n+1 i}]$) containing all $\mathbb{P}^1$'s making up the $1$-faces. The generic case corresponds to the collection of all these $\mathbb{P}^1$'s.

One proves this by induction, the essential reduction being that either $X$ is contained in the collection of hypersurfaces $x_1.x_2....x_{n+1}=0$ or $A$ is the twisted homogeneous coordinate ring of (ordinary commutative) $\mathbb{P}^n$ (and hence $X \simeq \mathbb{P}^n$).

Indeed, let $x_i$ act as a non-zero divisor on the point module $P$ then $P$ corresponds to a one-dimensional representation of the degree-zero part of the graded localization of $A$ at the normalizing element $x_i$. This degree zero part is a quantum polynomial ring in $n$ variables. Either these variables all commute in which case $A$ is the claimed twisted coordinate ring or one of the new variables $y_j = x_jx_i^{-1}$ vanishes on the one-dimensional representation and hence $x_j$ vanishes on $P$.

If $X \subset \mathbb{V}(x_1.x_2... x_{n+1})$ then one gets all point-modules on which $x_i$ vanishes by looking at the point-modules of the quotient $A/(x_i)$ which is again a quantum $\mathbb{P}^{n-1}$ and one gets the claimed $X$ by induction.

The induction starts with $n=1$ in which case $X=\mathbb{P}^1$ (as $A$ is a twisted coordinate ring) and $n=2$ in which case $X$ is either $\mathbb{P}^2$ or the triangle $x_1x_2x_3=0$. So for $n=3$ one has $X$ either isomorphic to $\mathbb{P}^3$ or the union of the point-modules of the $4$ quantum $\mathbb{P}^2$'s determined by the quotients $A/(x_i)$, each of these giving either a triangle or a $\mathbb{P}^2$. Etc.

  • $\begingroup$ Thank you for the detailed answer to my question. I am still on the way to completely understanding your argument, but this seems exactly what I was looking for. Do you know any reference where this example is studied in term of point modules? Is it well-known? $\endgroup$ – user2013 Oct 12 '12 at 4:46

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