I have been strongly recommended to read the book Naive Lie Theory. In the introduction one can read: "This naive approach to Lie theory is originally due to von Neumann, and it is now possible to streamline it by using standard results of undergraduate mathematics. To compensate for the limitations of the naive approach, end of chapter discussions introduce important results beyond those proved in the book, as part of an informal sketch of Lie theory and its history".

I would like to ask:

  • first of all, what is Lie theory and is there some agreement that it should be a fundamental part of undergraduate education?

  • could you elaborate a little on the book mentioned above? Is the naive approach somewhat too naive? Should I prefer other references?

  • $\begingroup$ Have you asked your TA directly why she/he strongly recommends Naive Lie Theory? $\endgroup$ – J W Dec 23 '14 at 7:27
  • $\begingroup$ @JW In his words, it's the only book accessible to an undergraduate. $\endgroup$ – Dal Dec 23 '14 at 9:28
  • $\begingroup$ Does your TA give other reasons for reading the book, apart from accessibility? Please note that I personally think that Naive Lie Theory is well worth reading (and the same goes for many of John Stillwell's books). However, I am reluctant to second-guess your TA. $\endgroup$ – J W Dec 23 '14 at 11:09
  • $\begingroup$ @JW Nope, otherwise I wouldn't be asking here. Anyway, thank you for sharing your opinion. Also, I've edited the question to make it more general. $\endgroup$ – Dal Dec 23 '14 at 11:15

is there some agreement that [Lie theory] should be a fundamental part of undergraduate education?


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    $\begingroup$ And you think it shouldn't be? $\endgroup$ – Dal Dec 25 '14 at 0:26
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    $\begingroup$ +1, though I agree that @Dal 's follow-up question is reasonable (and its answer would make for a much better response). $\endgroup$ – Benjamin Dickman Dec 26 '14 at 7:11

One refreshing aspect of Stillwell's book is that he takes the viewpoint not merely of von Neumann but also of Sophus Lie himself who thought of his algebras as groups of infinitesimal transformations, and Stillwell repeatedly returns to the intuitively appealing infinitesimal descriptions of the Lie algebra throughout the book.

Part of the reason this is called the naive approach is because infinitesimals used to be thought of as being naive.

  • $\begingroup$ Professor, would you add if you consider the study of Lie Theory important and why? $\endgroup$ – Dal Dec 28 '14 at 23:13

First of all let me mention that I'm a physics student and therefore cannot tell you whether Lie theory is interesting for a mathematician. Also I don't know the specific book you are referring to. However, I can tell you why Lie groups and Lie algebras are one of the most essential pieces of maths that a theoretical physicists needs.

Lie groups and algebras appear throughout physics to mathematically implement the concept of symmetry. For example, physics should be invariant under translations in space and in time, spatial rotations and boosts. Spatial rotations are described by the rotation ("special orthogonal") group SO(3), spacetime rotations, that is spatial rotations and boosts, are described by the bigger Lorentz group SO(3,1) and if you add the spacetime translations you end up with the Poincaré group. All of these groups are Lie groups. The Poincaré symmetry leads to the very way how you write down physical laws, using so-called Lorentz indices (covariant notation).

In quantum field theory and elementary particle physics you encounter even more Lie groups describing so called internal symmetries. For example, the internal symmetry group of the standard model describing the strong and electroweak interaction is SU(3)xSU(2)xU(1) where SU(n) and U(n) are again Lie groups. Such symmetries help you to write down the theory and also work with it, i.e. do the spontaneous symmetry breaking using the Higgs mechanism which leads to the breaking of an initial internal symmetry group to a smaller one. The so called grand unified theories for example start with a big group like SU(5) and break it down to the standard model group SU(3)xSU(2)xU(1) using the Higgs mechanism.

  • $\begingroup$ is representation theory a standard physics undergraduate course? (no, for shame I might add :) ) $\endgroup$ – James S. Cook Dec 25 '14 at 7:40
  • $\begingroup$ I guess, it depends on the university, in my university it isn't, unfortunately. I guess, this is because not every physics student intends to become theorist and for experimentalists it is indeed unnecessary. In my experience, however, group theory, especially Lie groups and algebras were the piece of maths I was missing most during the undergraduate study. $\endgroup$ – Photon Dec 25 '14 at 8:11
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    $\begingroup$ But, is it really unnecessary for experimentalists? I would offer that understanding basic group representations makes quantum mechanics much less mysterious. I thought physicists care about understanding physics. Even experimentalists. But, yes, the reality you experience is not unusual. The fact is that physics education is based more on tradition than on logic. Of course, the same could be said of the required education of many subjects. Maybe all. $\endgroup$ – James S. Cook Dec 25 '14 at 14:36

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