From Quantum Groups to Unitary Modular Tensor Categories

2005/03/11 by Eric C. Rowell, Rowell, Eric C. · 1 citation
Mathematics · #57R56 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Primary 20G42 #Quantum Algebra (math.QA) #Secondary 20F46 #math.CT #math.QA #msc:20F46 #msc:20G42 #msc:57R56

paper · pdf · doi:10.48550/arxiv.math/0503226

16 pages, final version to appear in Contemporary Mathematics. Version 2: minor typos fixed, references updated, Theorem 4.5 clarified. Version 3: major addition deriving generating functions for ranks, paper reorganized

openalex publication_date 2005/03/11 · arxiv created 2006/03/13 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Modular tensor categories are generalizations of the representation categories of quantum groups at roots of unity axiomatizing the properties necessary to produce 3-dimensional TQFTs. Although other constructions have since been found, quantum groups remain the most prolific source. Recently proposed applications to quantum computing have provided an impetus to understand and describe these examples as explicitly as possible, especially those that are "physically feasible." We survey the current status of the problem of producing unitary modular tensor categories from quantum groups, emphasizing explicit computations.

Citations

• Invariants of 3-manifolds via link polynomials and quantum groups
• From subfactors to categories and topology II: The quantum double of tensor categories and subfactors

Cited by

• Quantization via Linear homotopy types

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