Let $a_i \geq 2$ be natural numbers and $q_{ij}$ field elements of the field $k$ for $i>j$. A quantum complete intersection is the algebra $A:=k<x_1,...,x_n>/(x_i^{a_i},x_i x_j - q_{ij} x_j x_i)$. This is a local finite dimensional Frobenius algebra of dimension $a_1 a_2 ... a_n$.

Let $b_i$ be natural numbers with $0 \leq b_i \leq a_i -1$ and $c_i$ field elments. Define the vectors $b=(b_1,...,b_n)$ and $c=(c_1,...,c_n)$.

Define the modules (right ideals) by $M_c^b:=(c_1 x_1^{b_1}+...+c_n x_n^{b_n}) A$. So for example $M_c^b$ is equal to the algebra when choosing $b_i=0$ and $c_i=1$.

Specialising some parameters(I only looked at the at most three variables case), I could do some interesting contstructions but looking at the general case the computations seem to become a mess.

But maybe there are nice methods to answer the following questions:

What are those right ideals with dimension equal to half the dimension of the algebra? I think those build an especially nice subcategory, which is closed under syzygies. Reason: There is then a short exact sequence $0 \rightarrow U \rightarrow A \rightarrow $M_c^b$ \rightarrow 0$ with $U$ also having dimension equal to half the dimension of $A$. $U$ then should be equal to $uA$, when $u$ is an element such that $ (c_1 x_1^{b_1}+...+c_n x_n^{b_n}) u=0$.

What is the dimension of $Hom_A(M_c^b, M_{c'}^{b'})$? This should also answer when two such right modules are isomorphic. My guess is that they are isomorphic iff $b=b'$ and $c$ and $c'$ are equal in the respective projective space. at least when all $b_i \geq 1$. The special case $M_c^b=A$ would give the general dimension of $ M_{c'}^{b'}$.

When is $M_c^b$ a two-sided ideal?

Answers to 2. and 3. would also be interesting restricted to the right ideals with dimension equal to half the dimension of the algebra.

- (maybe very hard question) When is $A$ a Hopf algebra?