There's a welle known Hopf pairing between $SL_q(2)$ and $U_q(sl(2))$, I have no problem in construction such pairing wich can be easly be done by the coefficients of a 2-dimensional irreducible module on $U_q(sl(2))$.

Of course such a pairing implies the definition of a morphism from $U_q(sl(2))$ to $SL_q(2)^*$. This morphism will aloud all $SL_q(2)^*$ modules to become $U_q(sl(2))$ modules. Until here everything is fine.

Now I'm following Kassel, section VII.5 he want to proof that the irreducible modules on $\mathbb{K}_{q}^{n}\left[x,\,y\right]^*$ are isomorphic in fact to the irreducible one $V_{1,1}$ on $U_q(sl(2))$ of highest weight $q^n$. He does that showing that defining $$f(x^{i}y^{n-i})=\delta_{ni}$$, then you have that this functional act on every $u \in U_q(sl(2))$ as $$\\uf(x^{i}y^{n-i})=\left\langle u,\,\,a^{i}c^{n-i}\right\rangle $$ And then showing that is an highest weight vector.

I undersand the idea behind but I don't get why $$\\uf(x^{i}y^{n-i})=\left\langle u,\,\,a^{i}c^{n-i}\right\rangle $$ Can anybody explain me all the passages to get that?

Otherwise does anybody knows a better and or simpler way to proof the relation between irreducible module on $SL_q(2)^*$ and $U_q(sl(2))$? Thank you in advance


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