Why a group that describe rotations always have $su(2)$ Lie algebra?

Question

I'm reading the book Physics from Symmetry by Jakob Schwichtenberg. In part II the author explain the Lie group theory and in particular he treat the $SU(2)$ group. At a certain point the author tells us that $SU(2)$ is the covering group of the Lie algebra $su(2)$ and this involve that every group with that Lie algebra can be described using $SU(2)$. After that the author seems to take for granted that every group that describe 3D rotations must have the $su(2)$ Lie algebra and for this reason he study the $SU(2)$ representations. My question is, why every group that describe 3D rotations must have the $su(2)$ Lie algebra? Is it because $su(2$) Lie algebra encode some sort of behavior of the rotations that we assume to be true?

Answer 1

From the comments,

Yes is this the question, how do you know that there isn't another description of the rotations with another Lie algebra? because to me it looks like we have found 2 description of rotations, SU(2) and SO(3), we have seen that they have same Lie algebra and so we have assumed all the descriptions will have that Lie algebra.

I believe there is a fundamental misunderstanding here, so I'll try to quickly review the story.

I: Representations of SO(3)

The group $SO(3)$ is the rotation group in 3D space. It is defined by its fundamental representation as the $3\times 3$ real, orthogonal matrices with determinant $1$. This representation allows us to act on elements of $\mathbb R^3$ with elements of $SO(3)$ via standard matrix multiplication.

From here, it is natural to ask what effect a rotation has on something which isn't an element of $\mathbb R^3$. This leads us to the representation theory of Lie groups. Given some vector space $V$, we seek a map $\rho: SO(3) \rightarrow GL(V)$ (where $GL(V)$ is the set of invertible linear maps from $V\rightarrow V$) which has the following property:

$$\forall R_1,R_2 \in SO(3), \qquad \rho(R_1\circ R_2)= \rho(R_1)\circ \rho(R_2)$$

Such a map is called a representation of $SO(3)$ on the representation space $V$, and for every rotation $R\in SO(3)$, $\rho(R)$ provides the corresponding action on elements of $V$.

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II: Representations of $\frak{so}(3)$

As it turns out, it is quite cumbersome to work directly with representations of groups. Fortunately, we know that at least in some connected neighborhood of the identity element, we can form a one-to-one correspondence between a Lie group $G$ and its associated Lie algebra $\frak g$, with the elements of $G$ obtained from the elements of $\frak g$ by exponentiation. Therefore, rather than seeking representations of $SO(3)$ on vector spaces, we seek representations of $\frak{so}(3)$ on them, which we can then (maybe) exponentiate to obtain representations of the original group.

The fundamental representation of $\frak{so}(3)$ is the $3\times 3$ real, antisymmetric matrices. The standard basis $L_i$, $i=1,2,3$ has commutation relations $$[L_i,L_j]=\epsilon_{ijk}L_k$$

A representation of the Lie algebra $\frak{so}(3)$ on a vector space $V$ is a linear map $\varphi:{\frak{so}(3)}\rightarrow {\frak{gl}}(V)$ ( where ${\frak{gl}}(V)$ is the set of linear maps from $V\rightarrow V$) subject to the condition

$$\forall g,h \in {\frak{so}}(3), \qquad \varphi\big([g,h]\big) = \big[\varphi(g),\varphi(h)\big]$$

The linearity makes this much nicer to work with. We can simply search for sets of three matrices (or operators, in the infinite-dimensional case) which obey the right commutation relations, and this will constitute a representation of $\frak{so}(3)$.

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III: Projective Representations and the Universal Cover

It turns out that a representation of a Lie algebra $\frak g$ does not automatically yield a representation of the corresponding Lie group $G$ upon exponentiation if $G$ is not simply-connected (and $SO(3)$ is not). In this case, we encounter representations of $\frak{so}(3)$ which, when exponentiated, yield projective representations of $SO(3)$ instead; that is, we will have that

$$\forall R_1,R_2\in SO(3), \qquad \rho(R_1\circ R_2)=c(R_1,R_2) \rho(R_1)\circ\rho(R_2)$$

where $c(R_1,R_2)$ is some constant. Physically, this is actually desirable in the sense that Wigner's theorem tells us that we can represent symmetry transformations as unitary operators up to a phase. If we restricted our attention only to properly unitary representations of our symmetry group, then we would miss some. Unfortunately, projective representations can be a pain to work with because of these extra factors which we would need to keep track of.

This leads us to the concept of the universal cover. Given a Lie group $G$, its universal cover $U(G)$ is the unique simply connected Lie group which shares the same Lie algebra. Since it is simply connected, every representation of $\frak{g}$ gives rise to a genuine, non-projective representation of $U(G)$. In other words, rather than considering projective representations of $SO(3)$, we can consider genuine representations of $U\big(SO(3)\big)$.

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To summarize, our goal was to find representations of $SO(3)$ which can act on vector spaces other than $\mathbb R^3$. It is much more convenient to consider representations of the Lie algebra $\frak{so}(3)$, but because $SO(3)$ is not simply-connected, some representations of $\frak{so}(3)$ give rise to projective representations of $SO(3)$ rather than genuine ones.

In the context of quantum mechanics, this is actually a good thing, but projective representations are annoying to work with. Because every projective representation of $SO(3)$ is a genuine representation of the universal cover $U\big(SO(3)\big) \simeq SU(2)$, we can study the latter without worrying about those pesky factors we'd need to keep track of by studying the former.