# Generators of the Augmentation Ideal (Counit Kernel)

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?

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

• 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