QUANTUM GROUPS AT ROOTS OF UNITY AND MODULARITY

By S. Sawin. 2003

83 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/26b0ca36d2a24b4d1e4bc16842bec7ebbfd2d012

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Citations in the corpus

Braided Tensor Categories of Admissible Modules for Affine Lie Algebras (2017, background): “a b with coprime positive integers a,b such that ℓ is an admissible level for g. Let q = e 2πi 2d(ℓ+h∨) and Cℓ(g) be the semi-simpliﬁcation of the category of tilting modules of Uq(g) as described in [S]. Let Oℓ,ord be the braided tensor category constructed in the present paper. The ﬁrst conjecture”