Point modules of quantum projective space $\mathbb{P}^n$

Question

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.

Answer 1

$X$ 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.