# representation of SU(2)

The question is regarding SU(2) group and SU(2) algebra. The SU(2) group can be generated by exponentiating the generators of SU(2) algebra $X_a$ as $exp(i t_a X_a )$ with $t_a$ being three parameters. Genreally we use half the Pauli matrices as $X_a$ when discussing SU(2) algebra as well as the group. But if we use the spin-1 representation of $X_a$ (which can surely be done at the Lie Alebra level) can they be exponentiated to give another rep. of the SU(2) group (which would be 3 dimensional)?

• By the way, the group you obtain in this case is isomorphic to SO(3). If you choose infinitesimal rotations as the spin-1 representation, you obtain the SO(3) matrices directly. – The Vee Sep 20 '12 at 19:42

## 1 Answer

Yes. A representation of a Lie algebra always exponentiates to a representation of the simply connected Lie group corresponding to it. The representations of each are in bijective correspondence this way.

I recommend the free online book by Kirillov ( http://www.math.sunysb.edu/~kirillov/mat552/liegroups.pdf ) as a reference for this business.

• It exponentiates if the Lie algebra is finite-dimensional (as in case of SU(2)), but not necessarily in infinite dimensions. – Arnold Neumaier Sep 5 '12 at 8:40
• I second a vote for Kirillov's Lie Group notes! Download them and keep a copy lest they should disappear. The university sector's shrinking faster than a pierced balloon in my home land and, although Kirillov's present home land is not mine, it is also infected with the English language bacterium which seems to be an effective economic "rationalist" virus's vector organism! Also see my answer on how several different exponentiations can happen in finite dimensions, and I also recommend Chapter 8 of John Stillwell's "Naive Lie Theory". – WetSavannaAnimal Sep 27 '13 at 1:49