Let $M_q[2]$ be the algebra of quantum matrices over the complex numbers with the usual generators $a,b,c,d$ and the relations $ab = qba$, ... etc. Moreover, let $SL_q(2)$ be the quotient of $M_q(2)$ by the ideal generated by det$_q1$, where det$_q = ad  qbc$. Given a basis of $SL_q(2)$ it is easy to construct an embedding of $SL_q(2)$ into $M_q(2)$. What I would like to know is: Can anyone see a canonical way of embedding $SL_q(2)$ into $M_q(2)$?
