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For the Hopf algebra $SL_q(N)$ it is clear that the kernel of the counit contains the ideal generated by the elements $(u^i_i-1)$ and $u^i_j$, for $i \neq j$. However, I cannot seem to arrive at at proof that it is in fact generated by these elements. Does anyone how to do this?

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up vote 3 down vote accepted

Just observe that the quotient by the ideal generated by these elements is at most 1-dimensional.

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Why is that so? – Dyke Acland Dec 14 '10 at 23:05
.. and why does that suffice? – Dyke Acland Dec 14 '10 at 23:12
Let $I$ be the ideal of your quantum group $H$ generated by your guys. Every generator is scalar modulo $I$, thus $I$ is at most 1-dimensional. Bingo!! $I$ sits in the kernel of counit and you have inequality on codimensions that allows you to conclude that they are equal. – Bugs Bunny Dec 15 '10 at 7:19
Whooops, I meant to say that $H/I$ is at most 1-dimensional. – Bugs Bunny Dec 15 '10 at 7:20
Great! Thanks for that. I was trying to formulate a proof using the divisor theorem for polynomials, but your method is alot easier. – Dyke Acland Dec 15 '10 at 14:37

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