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 $M_q(2)$?

edited Jan 1 2011 at 22:25

Yemon Choi
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asked Jan 1 2011 at 17:12

John McCarthy
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The paper below doesn't quite answer your question, and yet... deepblue.lib.umich.edu/bitstream/2027.42/30793/1/… – Igor Rivin Jan 1 2011 at 23:23

Would you mind explaining what $q$ is? Also, could you give the relations besides $ab = qba$? – drbobmeister Jan 11 2011 at 21:50

It's OK I found Takeuchi's MSRI notes . . . – drbobmeister Jan 12 2011 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 2011 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 2011 at 0:19