Abstract
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 (sl2) 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 Sj(V) for V an irreducible representation of sl2. We obtain functional equations for the associated generating function. We also discuss ζ t (C q [S2]) for the standard q-deformed sphere.
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Acknowledgments
We thank Malek Abdesselam for pointing us to the classical Cayley–Sylvester formula used in Sect. 4. We also thank Thomas Prellberg for discussions and for computing the explicit formula after Lemma 4.6.
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Communicated by S.K. Jain.
The S. Majid was supported by a Senior Leverhulme Research Fellowship.
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Majid, S., Tomašić, I. On braided zeta functions. Bull. Math. Sci. 1, 379–396 (2011). https://doi.org/10.1007/s13373-011-0006-3
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DOI: https://doi.org/10.1007/s13373-011-0006-3
Keywords
- Riemann hypothesis
- Algebraic geometry
- Motivic zeta function
- Finite field
- Quantum groups
- q-Deformation
- Renormalisation
- Braided category