Narrow Results

We develop the basic representation theory of all quantum groups at all roots of unity (that is, for q any root of unity, where q is defined as in [18]), including Harish–Chandra's theorem, which allows us to show that an appropriate quotient of a subcategory gives a semisimple ribbon category. This work generalizes previous work on the foundations of representation theory of quantum groups at roots of unity which applied only to quantizations of the simplest groups, or to certain fractional levels, or only to the projective form of the group. The second half of this paper applies the representation theory to give a sequence of results crucial to applications in topology. In particular, for each compact, simple, simply-connected Lie group we show that at each integer level the quotient category is in fact modular (thus leading to a Topological Quantum Field Theory), we determine when at fractional levels the corresponding category is modular, and we give a quantum version of the Racah formula for the decomposition of the tensor product.

Let $q$ be a $2N$th root of unity where $N$ is odd. Let $U_q^s(s{l}_2)$ denote the quantum group with large center corresponding to the Lie algebra $s{l}_2ℂ$ with generators $E,F,K$, and $K^{-1}$. A semicyclic representation of $U_q^s(s{l}_2)$ is an $N$-dimensional irreducible representation $\rho :U_q^s(s{l}_2)\to M_N(ℂ)$, so that $\rho (E^N)=aId$ with $a\ne 0$, $\rho (F^N)=0$ and $\rho (K^N)=Id$. We construct a tangle functor for framed homogeneous tangles colored with semicyclic representations, and prove that for $(1,1)$-tangles coming from knots, the invariant defined by the tangle functor coincides with Kashaev’s invariant.

The Hennings invariant for the small quantum group associated to an arbitrary simple Lie algebra at a root of unity is shown to agree with the Witten–Reshetikhin–Turaev (WRT) three-manifold invariant arising from Chern–Simons field theory for the same Lie algebra and the same root of unity on all integer homology three-spheres, at roots of unity where both are defined. This partially generalizes the work of Chen et al. [On the relation between the WRT invariant and the Hennings invariant, Math. Proc. Cambridge Philos. Soc. 146(1) (2009) 151–163; Three-manifold invariants associated with restricted quantum groups, Math. Z. 272(3–4) (2012) 987–999] which relates the Hennings and WRT invariants for $\mathrm{\text{SL}}(2)$ and $\mathrm{\text{SO}}(3)$ for arbitrary rational homology three-spheres.

In this paper, we propose different notions of 𝔽_ζ-geometry, for ζ a root of unity, generalizing notions of 𝔽₁-geometry (geometry over the "field with one element") based on the behavior of the counting functions of points over finite fields, the Grothendieck class, and the notion of torification. We relate 𝔽_ζ-geometry to formal roots of Tate motives, and to functions in the Habiro ring, seen as counting functions of certain ind-varieties. We investigate the existence of 𝔽_ζ-structures in examples arising from general linear groups, matrix equations over finite fields, and some quantum modular forms.