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Edit: In physics, it seems that people usually study $su(2)$ but not only $sl(2)$? Why people study $su(2)$ but not only $sl(2)$?

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closed as off topic by Hans Lundmark, tomasz, rschwieb, Arkamis, Aang Sep 23 '12 at 17:29

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I think you should try to make this question less vague. –  Eric O. Korman Aug 16 '12 at 23:24
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Why do you think that people do not study $sl(2)$? Since that is quite false, it is somewhat difficult to answer your question! –  Mariano Suárez-Alvarez Aug 16 '12 at 23:26
    
(Your edit does not help... as it rendered the text more or less weird. In any case, both groups/lie algebras are studied!) –  Mariano Suárez-Alvarez Aug 16 '12 at 23:35
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Maybe you mean the role of $SU(2)$ in elementary particle physics, eg, Zizzi & Pessa: "The SU(2) gauge theory was the first non-abelian generalization of the U(1) gauge theory of electromagnetism. It was introduced by Yang and Mills in 1954 in order to extend the SU(2) global invariance of isotopic symmetry to a local SU(2) invariance. This requires the introduction of three vector fields, one for each generator of SU(2)." / arxiv.org/pdf/1104.0114v1.pdf –  alancalvitti Aug 16 '12 at 23:44

2 Answers 2

up vote 2 down vote accepted

The form of the question is somewhat iffy, but may be intended to ask a question that has a sense that I understand: since most representations are on complex vector spaces, a representation of a Lie algebra is indistinguishable from a representation of its complexification. The complexifications of su(2) and sl(2,R) are both sl(2,C)... often unhelpfully written merely as sl(2), not acknowledging the real-versus-complex game.

Perhaps another iteration of the question would make clearer what other/remaining issues might be...

Edit: The Lie groups $SU(2)$ and $SL_2(\mathbb R)$ are easier to distinguish: the former is compact and the latter is not, which has as consequence that the irreducible unitary repns of the former are all finite-dimensional (indexed by highest weight), while the irreducible unitaries of $SL_2(\mathbb R)$ are all infinite-dimensional (apart from the trivial repn). Appreciating this point about the Lie groups is probably necessary in order to appreciate distinctions about the Lie algebras which otherwise might seem capricious.

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thank you. What are the differences between $su(2)$ and $sl(2, R)$? –  LJR Aug 17 '12 at 15:29

I can try to answer. Elements of $SL(n)$ preserve only volume, but elements of $SU(n)$ also preserve distance.

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