# $\mathrm{SU}(2)$ as a representation of the rotation group

I have read in a book that the group $$\mathrm{SU}(2)$$ is one of the irreducible representations of the rotation group. The book begin saying that the rotation group has 3 generators $$J_{1}, J_{2}$$ and $$J_{3}$$, and the algebra is $$[J_{i},J_{j}] = i \epsilon _{ijk} J_{k}.$$ After, considering the Casimir operator $$J^{2} = J_{1}^{2} + J_{2}^{2} + J_{3}^{2}$$ the book finds the eigenvalues of $$J^{2}$$ and $$J_{3}$$ in the same basis: $$J^{2} |j,m \rangle = j(j + 1) |j,m \rangle ,$$ $$J_{3} |j,m \rangle = m |j,m \rangle ,$$ where $$j = 0, 1/2, 1, 3/2, \ldots$$ and $$m = -j, -j+1,\ldots, 0, \ldots , j-1, j$$. It says that for each value of $$j$$ corresponds one irreducible representation, for $$j = 1/2$$ this representation is the group $$\mathrm{SU}(2)$$, for $$j = 1$$ this is the $$\mathrm{SO}(3)$$, and so on...

But the book doesn't prove that this is in fact a representation. How can I do this?

And what is this rotation group? How is it defined?

Edit: I have read it in the book Matemática para físicos com aplicações by João Neto, vol. 1 (the book is in Portuguese). The book says "when two groups have the same algebra, we say that these are not two distinct groups, but different representations of the same group". So he concluded that $$\mathrm{SU}(2)$$ and are $$\mathrm{SO}(3)$$ are representations of the same group, which it call of rotation group.

• May 19, 2019 at 21:03
• I wonder if the statement should be that the Lie algebra $\hbox{su}(2)$ is a representation of the rotation group $SO(3)$. May 19, 2019 at 21:39
• @user52817 It should not. A Lie algebra and a group representation are two different things. As Emilio Pisanty explains, the relationship is that SU(2) is the “covering group” of SO(3). Both are groups. Neither is a Lie algebra nor a group representation. Both have Lie algebras and group representations. May 19, 2019 at 22:50
• @G.Smith My guess is that user52817 was referring to the fact that the structure constants do in fact give a representation, viz. the adjoint. (But this is a rep. of the algebra, not the group, so the claim is indeed wrong). May 20, 2019 at 0:06
• @AccidentalFourierTransform It isn’t kosher to consider the structure constants of a Lie algebra to “be” the Lie algebra itself, is it? May 20, 2019 at 0:10

when two groups have the same algebra, we say that these are not two distinct groups, but different representations of the same group

This is bonkers. When two groups have the same algebra, then we say that they have the same algebra, i.e. that they are locally isomorphic. We do not say that they are the same group; we only say this when there is a group isomorphism between the two, and this is not the case with $$\rm SU(2)$$ and $$\rm SO(3)$$.

Moreover, that quote (if the translation is accurate) is abusing its notation to a pretty remarkable degree, particularly as regards the term "representation". Within this context, the term "representation" of a group $$G$$ should only be used in this specific technical sense, i.e. to refer to a vector space $$V$$ and to the linear action $$R:G\to \mathrm{End}(V)$$ on it.

The author seems to be getting at some form of "sameness" between $$\rm SU(2)$$ and $$\rm SO(3)$$, which does exist:

• the two groups are locally isomorphic, and
• more globally, $$\rm SU(2)$$ is the covering group for $$\rm SO(3)$$.

However, there is no standard sense in which the two can be considered "representations" of some other, "more abstract" group. If your translations are accurate, then the author is drawing outside the lines. It's not quite serious enough that I would tell you to drop the book immediately and find one which isn't wrong, but it's definitely a strong warning sign $$-$$ take everything on that book with a grain of salt, and start looking for a good replacement.

• Regarding the last paragraph (v1), I have found such statements elsewhere. For the sake of clarity, do you consider $\mathrm{SU}(2)$ and $\mathrm{SO}(3)$ as matrix Lie groups? Dec 9, 2022 at 19:25
• @Fey You're gonna have to define what you mean by "matrix Lie group" for me, I'm afraid. I don't see much use for the term. Dec 10, 2022 at 9:36
• I should point out first that I agree with the paragraph I have quoted. A matrix Lie group is a Lie group whose elements are matrices (more specifically a subgroup of $\mathrm{GL}(n;\mathbb{C})$, cft eg. Hall). For example, $\mathrm{SO}(3)$ would be defined as the group of $3\times3$ orthogonal matrices with unit determinant. In that case, the only way I would try to make sense of considering such group as a "representation" would be as the image of a representation of the rotation group you refer to at the beginning of this answer. Dec 10, 2022 at 10:50
• @Feynman_00 So long as you're not invoking any non-compact Lie groups, there is no difference between "matrix" Lie groups and plain-vanilla Lie groups. This is because any compact Lie group is isomorphic to a subgroup of $\mathrm{GL}(n;\mathbb C)$ for some $n$. Whether you "define" the group to be something else is immaterial and it makes no practical difference -- there is an isomorphism to a matrix group. (Hence the claim that there isn't much use for the term.) Dec 13, 2022 at 20:10
• As regards the word "representation", here the OP's textbook is quite clearly using the term improperly and away from its standard meaning when groups are concerned, so anything in the text of this answer needs to be understood as addressing that misuse of language. Dec 13, 2022 at 20:13