Embedding Quantum SL(2) into the Quantum Matrices

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$_q-1$, 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 $M_q(2)$?

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The paper below doesn't quite answer your question, and yet... deepblue.lib.umich.edu/bitstream/2027.42/30793/1/0000447.pdf –  Igor Rivin Jan 1 '11 at 23:23
Would you mind explaining what $q$ is? Also, could you give the relations besides $ab = qba$? –  drbobmeister Jan 11 '11 at 21:50
It's OK I found Takeuchi's MSRI notes . . . –  drbobmeister Jan 12 '11 at 7:43
The relations are very well known. See, for example, Majid's "What is a Quantum Group" linked on the Wikipedia Quantum Group page. –  John McCarthy Jan 12 '11 at 13:21
I'm not quite sure what you are looking for here. Could you be more explicit about canonical? Sorry if I'm being slow. –  B. Bischof Nov 13 '11 at 0:19