On braided zeta functions

We propose a ribbon braided category approach to zeta-functions in q-deformed geometry. As a proof of concept we compute ${\zeta }_t(C^n)$ where $C^n$ is viewed as the standard representation in the category of modules of $U_q(s{l}_n)$. We show that the same ${\zeta }_t(C^n)$ is obtained for the $n$-dimensional representation in the category of $U_q(s{l}_2)$ modules. We show that this implies and is equivalent to the generating function for the decomposition into irreducibles of the symmetric tensor products $S^j(V)$ for $V$ an irreducible representation of $s{l}_2$. We discuss ${\zeta }_t(C_q(S^2))$ for the standard q-deformed sphere.