Frobenius algebras, Hopf algebras and 3-categories PDF
Preview Frobenius algebras, Hopf algebras and 3-categories
Frobenius algebras, Hopf algebras and 3-categories David Reutter University of Oxford Hopf algebras in Kitaev’s quantum double models Perimeter Institute, Canada August 3, 2017 DavidReutter Hopfalgebrasand3-categories August3,2017 1/34 hand-wavy-ness t im e The plan Part 1. Motivation Part 2. 2-categories Part 3. 3-categories Part 4. Hopf algebras Part 5. Higher linear algebra Part 6. Lattice models DavidReutter Hopfalgebrasand3-categories August3,2017 2/34 The plan hand-wavy-ness Part 1. Motivation Part 2. 2-categories Part 3. 3-categories t im Part 4. Hopf algebras e Part 5. Higher linear algebra Part 6. Lattice models DavidReutter Hopfalgebrasand3-categories August3,2017 2/34 The plan Part 6. Lattice models Part 5. Higher linear algebra Part 4. Hopf algebras e m Part 3. 3-categories ti Part 2. 2-categories Part 1. Motivation hand-wavy-ness DavidReutter Hopfalgebrasand3-categories August3,2017 2/34 Part 1 Motivation DavidReutter Hopfalgebrasand3-categories August3,2017 3/34 Higher algebra lets us compose in higher dimensions: L (cid:15) M η N What is higher algebra? Ordinary algebra lets us compose along a line: xy2zyx3 DavidReutter Hopfalgebrasand3-categories August3,2017 4/34 What is higher algebra? Ordinary algebra lets us compose along a line: xy2zyx3 Higher algebra lets us compose in higher dimensions: L (cid:15) M η N DavidReutter Hopfalgebrasand3-categories August3,2017 4/34 s t u d y topology in termsofalgebra = = a harder a simpler + + topology algebra topology algebra ‘outsource’ algebra to topolo g y Frobenius algebras as a ‘shadow’ of a two dimensional theory. = = Next hour: Hopf algebras as a ‘shadow’ of a three dimensional theory. A tradeoff between algebra and topology DavidReutter Hopfalgebrasand3-categories August3,2017 5/34 as a ‘shadow’ of a two dimensional theory. s t u d y topology in termsofalgebra = = a harder a simpler + + topology algebra topology algebra ‘outsource’ algebra to topolo g y Next hour: Hopf algebras as a ‘shadow’ of a three dimensional theory. A tradeoff between algebra and topology Frobenius algebras = = DavidReutter Hopfalgebrasand3-categories August3,2017 5/34 s t u d y topology in termsofalgebra a harder a simpler + + topology algebra topology algebra ‘outsource’ algebra to topolo g y Next hour: Hopf algebras as a ‘shadow’ of a three dimensional theory. A tradeoff between algebra and topology Frobenius algebras as a ‘shadow’ of a two dimensional theory. = = = = DavidReutter Hopfalgebrasand3-categories August3,2017 5/34
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