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Is Abstract Algebra useful in theoretical Relativity and/or Cosmology? If so can anyone give me some examples or point me towards a good book with that emphasis if it is one?

Thanks in advance.

aortizmena
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    I think you need to be more specific about the aspect of modern algebra (abstract algebra?) in which you are interested. The term covers far too vast a range of topics to be enumerated in a short space. The short answer is, yes, algebra is extremely useful. Particularly that of Lie groups. – kleingordon Mar 27 '12 at 05:16
  • Yes, I'm thinking about abstract algebra, group theory, ring theory, Galois Theory, Representaion theory, etc. But the thing is I'm just taking an introductory course in Abstract Algebra so my experience with that branch of mathematics is very limited. – aortizmena Mar 27 '12 at 05:23
  • Related: http://physics.stackexchange.com/questions/6108/comprehensive-book-on-group-theory-for-physicists . Have you found the answers there insufficiently helpful? The references will certainly touch on applications to relativity. – kleingordon Mar 27 '12 at 07:25
  • I know Modern Algebra has lots of applications to quantum mechanics, cristalography, particle physics. My question goes more towards the importance of modern algebra in relativity and/or cosmology in partucular. Thanks. – aortizmena Mar 27 '12 at 07:52
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    Google search: https://www.google.com/search?ix=uh&sourceid=chrome&ie=UTF-8&q=group+theory+relativity – Manishearth Mar 27 '12 at 11:45
  • @aortizmena: the term 'Modern Algebra' is not often used in English (perhaps you are translating it from another language?). The subjects you mentioned tend to come under the term 'abstract algebra'. – genneth Mar 27 '12 at 14:17
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    @aortizmena: "modern algebra" is discrete rings, finite group theory, geometric (discrete) group theory, and various algebras, and has limited contact with physics. Only lie algebras and division algebras show up regularly. The closest is in string theory, and even there, the group theory tends to be on the simple side. The only physics application I know for a nontrivial algebraic construction is the Hopf algebra renormalization of Connes and Kreimer, but you should learn the stuff anyway. Good motivations come from equation solving and computational applications, not physics. – Ron Maimon Mar 28 '12 at 08:01

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Modern Algebra is used quite a bit in Quantum Field Theory to represent the Lorentz Group.

Since successive boosts can be performed in one boost, we can see that a boost applied to boost is a boost, implying that Lorentz Boosts form a group. You then use some representation theory to represent those boosts as matrices with some basis, and you can apply these to fields to boost them into different reference frames.

For relativity and cosmology, it's mostly differential geometry you'll need, but as vector spaces are also groups, I suppose you could say Modern Algebra is involved there.

P O'Conbhui
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One of the formulations of QFT relies on the algebra (to be specific -- a CCR algebra) of creation and annihilation operators.

Kostya
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