Hello, for my bachelor's thesis I need to understand the Hopf Algebra of Feynman Diagrams. As I have only litte knowledge in Algebra by now I wanted to ask where I could start and what preknowledge I should have in order to understand this topic. Can you perhaps recommend a textbook or some other reference regarding Hopf Algebra or Quantum Groups suitable for my need? Thank you in advance.
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$\begingroup$ For a physicist with an interest in QED perhaps a good place to start is Sec. 3.3 Fusion, Splitting, and Hopf Algebras in E. Zeidler's book Quantum Field Theory II Quantum Electrodynamics. $\endgroup$– Tom CopelandJan 9, 2013 at 23:12
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$\begingroup$ For more intros to Hopf algebras in physics, see arXiv papers jointly authored by Blasiak, Duchamp, Horzela, Penson and Solomon, and papers by Figueroa and Gracia-Bondi. $\endgroup$– Tom CopelandAug 19, 2015 at 19:42
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2 Answers
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Dominique Manchon's lecture notes, which are very well-known amongst people working on Connes--Kreimer renormalisation, offer exactly the sort of detailed, accessible introduction to Hopf algebras and Connes--Kreimer renormalisation that you're looking for. However, you should first be thoroughly comfortable with abstract linear algebra and with the basics of ring and module theory, and you should be familiar with the basic language of category theory and of representation theory. Roughly speaking, if you can follow along with Chapter 1 of the notes, you should have the bare minimum needed.
answered Jan 9, 2013 at 17:55
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I originally found the motivation for the definition of a Hopf algebra difficult to see. This article by Scott Carnahan makes the definition seem like a completely natural generalisation of the definition of a group. Think of that article as a "pre-introduction" that makes other introductions to Hopf algebras easier to grasp.
answered Jan 9, 2013 at 21:10