

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer

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

share|cite|improve this question

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


I think you should try to make this question less vague. – Eric O. Korman Aug 16 '12 at 23:24
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
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)." / – alancalvitti Aug 16 '12 at 23:44
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.

share|cite|improve this answer
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.

share|cite|improve this answer
