How to compute the inverse of a quantum determinant? Let $X=(x_{ij})_{mn}$ be a quantum matrix with the commutation relations between entries:
\begin{alignat*}{2}
& x_{ij} x_{il}  = q  x_{il} x_{ij},  && j < l, \\
& x_{ij} x_{kj}  = q x_{kj} x_{ij}, && i<k, \\
& x_{ij} x_{kl} = x_{kl} x_{ij}, && i<k, j>l, \\
& x_{ij} x_{kl} = x_{kl} x_{ij} + (q - q^{-1})x_{il}x_{kj}, \quad  && i < k, j < l.
\end{alignat*}
For $n \in \mathbb{Z}_{>0}$, denote $[n]=\{1, \ldots, n\}$. For any $I \subset [m]$, $J \subset [n]$, $|I|=|J|=l \in \mathbb{Z}_{>0}$, a quantum minor $\Delta_{I,J}$ is defined as follows
\begin{align*}
\Delta_{I,J} = \sum_{\sigma \in S_l} (-q)^{\ell(\sigma)} x_{i_1, j_{\sigma(1)}} \cdots x_{i_l, j_{\sigma(l)}},
\end{align*}
where $\{i_1 < \cdots < i_l\} = I$, $\{j_1< \cdots < j_l\}=J$, and $\ell(\sigma)$ is the length of the permutation $\sigma$.
When $m=n$ and $I=[n]$, $\Delta_{I,I}=\det_q(X)$ is the quantum determinant of $X$. How to compute $(\det_q(X))^{-1}$. Are there some references about this? Thank you very much.
 A: Algebraic properties of Manin matrices, page 8: $(\det_q M)^{-1}=\det_q(M^{-1})$.
A: The determinant $\text{det}_q(X)$ is not an invertible element of the algebra $X$. To see this look at the case of $X$, for $n=2$, and try to formulate an argument in terms of the standard basis 
$$
X_{11}^a X_{12}^b X_{21}^c X_{22}^d.
$$
Think of the $q=1$ case, where the algebra $X$ reduces to the coordinate algebra of $M_2(\mathbb{C})$. The determinant function will vanish on all matrices  with trivial determinant, and so,  cannot admit an inverse/
For $\text{det}$ to be invertible, you need to restrict to GL$(\mathbb{C})$. Dually this corresponds to adjoining an inverse to $\text{det}$. In the $q$-deformed case we mimic this and construct the quantum general linear group by adjoining a inverse. Explicitly, since $det_q(X)$ is well-known to be a central element of $X$, it is easy to attach an inverse viewed as an Ore extension, or equivalently just add  an extra  generator $\text{det}_q$ along with the obvious additional relations. 
The quantum special linear group can then be constructed from the quantum general linear group by quotienting by the ideal generated by $\text{det}_q(X) - 1$. Alternatively, one can go straight to the quantum special linear group by quotienting $X$ by the ideal generated by $\text{det}_q(X) - 1$.
This is of course closely related to the fact $X$ does not admit a Hopf algebra structure, but the 
quantum general linear, and special, groups do. This is all explained in Klimyk and Schmudgen's text, for example.
