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I read a textbook on Quantum Groups by Kassel, and did not understand the following proof of well-definedness of $\Delta$ (comult.) and $\epsilon$ (counit) on $GL_q(2)$ and $SL_q(2)$: (pg 84)

The proof is sketchy and describes that to prove well-definedness on $SL_q(2)$, it suffices to show $$\Delta (det_q -1)=(det_q -1)\otimes det_q + 1\otimes (det_q -1)$$ (how?)

and $$\epsilon (det_q -1)=0$$

The part I don't understand is how to get the above two equations and how does that prove well-definedness.


Additional theorems that may help which are defined earlier are $\Delta (det_q)=det_q \otimes det_q$ and $\epsilon (det_q)=1$.

Sincere thanks for any help.

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do you understand how the bialgebra structure on $M_q(2)$ works? –  mebassett Dec 17 '12 at 13:26
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1 Answer 1

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First, note that $SL_q(2)$ remains an algebra, because $M_q(2)$ is an algebra, and imposing the condition that $\text{det}_q = 1$ is the same as factoring out by the ideal generated by all multiples of $\text{det}_q -1$ (denoted by $I=(\text{det}_q-1)$. That it remains a coalgebra follows from the fact that $I$ is also a coideal, that is

$$\Delta(I) \subset SL_q(2) \otimes I + I \otimes SL_q(2)$$

We see that this is true from Kassel's computation of $\Delta(\text{det}_q -1)$. To get it, he just does the "add a copy, subtract a copy" trick. You should already know that $\Delta(\text{det}_q) = \text{det}_q \otimes \text{det}_q$. From the fact that the coproduct is linear you then have

$$ \Delta(\text{det}_q -1) = \text{det}_q \otimes \text{det}_q - 1 \otimes 1 $$

Doing the "adding and subtract a copy trick" we have

$$ \Delta(\text{det}_q) = \text{det}_q \otimes \text{det}_q - 1 \otimes \text{det}_q + 1 \otimes \text{det}_q - 1 \otimes 1$$

Then grouping like tensor products gives us Kassel's computation.

Since the coproduct was an algebra morphism before we passed to the quotient algebra, it remains an algebra morphism afterwards. You can verify this by writing down the appropriate commutative diagrams for $M_q(2)$ and $SL_q(2)$ and then linking them up with the canonical projection of $M_q(2)$ to its quotient. (If you need to verify this, I'd suggest you do it generally, with an any hopf algebra $H$ and quotient $H/I$ for a co/ideal $I$).

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Sincere thanks! I don't get how Kassel computes $\Delta (det_q-1)$ though. Also, I don't really understand coideals. (my background is undergraduate, with highest level course being introductory galois theory) –  yoyostein Dec 17 '12 at 16:26
    
I re-wrote it a bit more coherently, also describing coideals a tad bit more and explaining how he gets the computation. You should ensure you have a good idea of ideals and quotient rings if you don't have that already. coideals were described in ch3. I'm a 2nd year phd student and I find this hopelessly confusing myself. regards! –  mebassett Dec 17 '12 at 22:09
    
a word of warning about kassel - it's a great book, but he tends to be a bit "over axiomatic" and it's a bit hard to see the forest through the trees sometimes. For instance, it's a lot easier to just compute $\Delta(\text{det_q})$ then to follow his chain of propositions. It helps to have a few other texts handy to cross-reference these things. –  mebassett Dec 17 '12 at 22:15
    
thanks a lot! what other texts do you recommend? –  yoyostein Dec 24 '12 at 13:19
    
I'm working through majid's "primer on quantum groups" (disclaimer: majid is my phd supervisor) along with kassel, and I have a few other texts handy, but they're harder for an undegrad to crack. feel free to email me about it, though. –  mebassett Dec 28 '12 at 15:11
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