On braided zeta functions

We propose a braided approach to zeta-functions in q-deformed geometry, defining ζ t for any rigid object in a ribbon braided category. We compute ${\zeta_t(\mathbb{C}^n)}$ where ${\mathbb{C}^n}$ is viewed as the standard representation in the category of modules of U q (sl n ) and q is generic. We show that this coincides with ${\zeta_t(\mathbb{C}^n)}$ where ${\mathbb{C}^n}$ is the n-dimensional representation in the category of U q (sl 2) modules and that this equality of the two braided zeta functions is equivalent to the classical Cayley–Sylvester formula for the decomposition into irreducibles of the symmetric tensor products S j (V) for V an irreducible representation of sl 2. We obtain functional equations for the associated generating function. We also discuss ζ t (C q [S 2]) for the standard q-deformed sphere.