I am more interested in this question: given $\{ a, b\} $ in $k_q[x, y]$ where $q^n = 1$, what are the characteristics of such pair $\{ a, b\}$?. This question is much easier in the commutative case, that is, the polynomial ring with two variables, where we can say two elements $\{ f, g\}$ in $k[x, y]$ are regular sequence if and only if gcd$(f, g) = 1$ if and only if $\dim_k k[x, y]/(f, g) < \infty$. I am expecting an analogue in the noncommutative case, more specifically, in the quantum plane $k_q[x, y]$ where $q$ is not generic.

The result I referred to is as follows: Let $A$ be locally finite graded $k$-algebra. Let $x_1, \ldots, x_n$ be a normal sequence of homogeneous elements with deg$(x_i) = d_i$. Then $\{ x_1, \ldots, x_n\}$ is a regular sequence if and only if the Hilbert polynomials of $A$ and $A/(x_1, \ldots, x_n)$ satisfy $$ H_{A/(x_1, \ldots, x_n)} (t) = \Pi_{i=1}^n (1 - t^{d_i}) H_A(t).$$