Quantum determinant ($D_q=ad-qbc$)

For the two-by-two RTT construction with the standard Yang-Baxter R-matrix, the displayed determinant commutes with the matrix generators and satisfies $\Delta D_q=D_q\otimes D_q$ and $\epsilon (D_q)=1$. Imposing $D_q=1$ gives the coordinate Hopf algebra of quantum SL2; adjoining its inverse instead gives the quantum general-linear coordinate algebra.