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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)?

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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. – Vašek Potoček Sep 20 '12 at 19:42

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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.

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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

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